Textbook Of Vector Analysis And Coordinate Geometry 9350843145, 9789350843147

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Textbook of Vector Analysis and Coordinate Geometry

Textbook of Vector Analysis and Coordinate Geometry

Dr. Ramanand Singh

CENTRUM PRESS NEW DELHI-110002 (INDIA)

CENTRUM PRESS H.O.: 4360/4, Ansari Road, Daryaganj, New Delhi-110002 (India) Tel.: 23278000, 23261597, 23255577, 23286875 B.O.: No. 1015, Ist Main Road, BSK IIIrd Stage IIIrd Phase, IIIrd Block, Bengaluru-560 085 (India) Tel.: 080-41723429 Email: [email protected] Visit us at: www.centrumpress.com

Textbook of Vector Analysis and Coordinate Geometry © Reserved First Edition, 2014 ISBN 978-93-5084-314-7

PRINTED IN INDIA

Printed at AnVi Composers, New Delhi

Contents Preface 1. Introduction to Vector Analysis 2. Coordinate Expressions 3. Continuity Equation 4. Multipole Expansion 5. Solenoidal Vector Field 6. Coordinate System 7. Affine Coordinate System 8. Uniqueness of Polar Coordinates Bibliography Index

Preface Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning. In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes’ numerical definitions and representations. The numerical output, however, might also be a vector or a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Vector analysis, a branch of mathematics that deals with quantities that have both magnitude and direction. Some physical and geometric quantities, called scalars, can be fully defined by specifying their magnitude in suitable units of measure. Thus, mass can be expressed in grams, temperature in degrees on some scale, and time in seconds. Scalars can be represented graphically by points on some numerical scale such as a clock or thermometer. There also are quantities, called vectors, that require the specification of direction as well as magnitude. Velocity, force, and displacement are examples of vectors. A vector quantity can be represented graphically by a directed line segment, symbolized by an arrow pointing in the direction of the vector quantity, with the length of the segment representing the magnitude of the vector. A prototype of a vector is a directed line segment AB that can be thought to represent the displacement of a particle from its initial position A to a new position B. To distinguish vectors from scalars it is customary to denote vectors by boldface letters. Thus the vector AB in Figure 1 can be denoted by a and its length (or magnitude) by |a|. In many problems the location of the initial point of a vector is immaterial, so that two vectors are regarded as equal if they have the same length and the same direction. In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in ‘the xcoordinate’. The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry. Presentation of matter in this textbook is very simple and lucid and has covered wide

variety of problems, solved and unsolved both. This book will also provide great help to those who are preparing for IAS/PCS and various other competitive examinations which demands coordinate geometry of two and three dimension and vector calculus in one book. — Editor

Chapter 1: Introduction to Vector Analysis Vector Calculus Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3. The term “vector calculus” is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below. Basic Objects The basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valued functions). These are then combined or transformed under various operations, and integrated. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below. Vector Operations Algebraic Operations: The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of:

Scalar Multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. The term “scalar” itself derives from this usage: a scalar is that which scales vectors. Scalar multiplication is different from the scalar product, which is an inner product between two

vectors. Definition In general, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is denoted cv. Scalar multiplication obeys the following rules (vector in boldface): • Left distributivity: (c + d)v = cv + dv; • Right distributivity: c(v + w) = cv + cw; • Associativity: (cd)v = c(dv); • Multiplying by 1 does not change a vector: 1v = v; • Multiplying by 0 gives the null vector: 0v = 0; • Multiplying by -1 gives the additive inverse: (-1)v = -v. Here + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field. Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector. As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is Kn, then scalar multiplication is defined component-wise. The same idea goes through with no change if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, then the only change is that the order of the multiplication may be reversed, resulting in the distinct operations left scalar multiplication cv and right scalar multiplication vc.

Euclidean Vector This article is about the vectors mainly used in physics and engineering to represent directed quantities. In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or - as here - simply a vector) is a geometric object that has a magnitude (or length) and direction and can be added according to the parallelogram law of addition. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe

physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors. It is important to distinguish Euclidean vectors from the more general concept in linear algebra of vectors as elements of a vector space. General vectors in this sense are fixed-size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be real numbers, and the normal Euclidean concepts of length, distance and angle may not be applicable. (A vector space with a definition of these concepts is called an inner product space.) In turn, both of these definitions of vector should be distinguished from the statistical concept of a random vector. The individual items in a random vector are individual real-valued random variables, and are often manipulated using the same sort of mathematical vector and matrix operations that apply to the other types of vectors, but otherwise usually behave more like collections of individual values. Concepts of length, distance and angle do not normally apply to these vectors, either; rather, what links the values together is the potential correlations among them. Overview A vector is a geometric entity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction. In rigorous mathematical treatments, a vector is defined as a directed line segment, or arrow, in a Euclidean space. When it becomes necessary to distinguish it from vectors as defined elsewhere, this is sometimes referred to as a geometric, spatial, or Euclidean vector. As an arrow in Euclidean space, a vector possesses a definite initial point and terminal point. Such a vector is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a and in space represent the same free vector if they have free vector. Thus two arrows the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB2 A2 is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions and to more formal approaches with much wider applications. Examples in One Dimension Since the physicist’s concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons. If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F is -15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement As of 4 metres to the right would be 4 m or -4 m, and its magnitude would be 4 m regardless. In Physics and Engineering Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is

speed. For example, the velocity 5 metres per second upward could be represented by the vector (0,5) (in 2 dimensions with the positive y axis as ‘up’). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field. In Cartesian Space In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points A = (1,0,0) and B = pointing from the point x=1 on the x-axis to the (0,1,0) in space determine the free vector point y=1 on the y-axis. Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin O = (0,0,0). Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin up the positive x-axis. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. For example, the sum of the vectors (1,2,3) and (-2,0,4) is the vector (1, 2, 3) + (-2, 0, 4) = (1 - 2, 2 + 0, 3 + 4) = (-1, 2, 7). Euclidean and Affine Vectors In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, a length or magnitude and a direction to vectors. In turn, the notion of direction is strictly associated with the notion of an angle between two vectors. When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length (the square root of the dot product of a vector by itself) and angle (a function of the dot product between any two vectors). In three dimensions, it is further possible to define a cross product which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of the parallelogram). However, it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors). An important example is Minkowski space that is important to our understanding of special relativity, where there is a generalization of length that permits non-zero vectors to have zero length. Other physical examples come from thermodynamics, where many of the quantities of interest can be considered vectors in a space with no notion of length or angle. Generalizations In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on some auxiliary coordinate system or reference frame. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the

components of the vector (or measurements taken with respect to the reference frame) must change to compensate. The vector is called covariant or contravariant depending on how the transformation of the vector’s components is related to the transformation of coordinates. In general, contravariant vectors are “regular vectors” with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as gradient. If you change units (a special case of a change of coordinates) from metres to millimetres, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm–a contravariant change in numerical value. In contrast, a gradient of 1 K/m becomes 0.001 K/mm–a covariant change in value. Tensors are another type of quantity that behave in this way; in fact a vector is a special type of tensor. In pure mathematics, a vector is any element of a vector space over some field and is often represented as a coordinate vector. The vectors described in this article are a very special case of this general definition because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has “magnitude and direction”. History The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. The immediate predecessor of vectors were quaternions, devised by William Rowan Hamilton in 1843 as a generalization of complex numbers. His search was for a formalism to enable the analysis of three-dimensional space in the same way that complex numbers had enabled analysis of twodimensional space. In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called “scalar” and “vector”: The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion. —W. R. Hamilton, London, Edinburgh & Dublin Philosophical Magazine 3rd series 29 27 (1846) Whereas complex numbers have one number (i) whose square is negative one, quaternions have three independent such numbers (i, j, k). Multiplication of these numbers by each other is not commutative, e.g., ij = – ji = k. Multiplication of two quaternions yields a third quaternion whose scalar part is the negative of the modern dot product and whose vector part is the modern cross product. Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator and is very close to modern vector analysis. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwell’s Treatise on Electricity and Magnetism, separated off their vector part for independent treatment. The first half of Gibbs’s Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis.

Representations Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, a as a. (Uppercase letters are typically used to represent matrices.) Other conventions include or a, especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B, it can also be as or AB. denoted

Vectors are usually shown in graphs or other diagrams as arrows (directed line segments). Here the point A is called the origin, tail, base, or initial point; point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector’s magnitude, while the direction in which the arrow points indicates the vector’s direction.

On a two-dimensional diagram, sometimes a vector perpendicular to the plane of the diagram is desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ©) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the vanes of an arrow from the back.

Figure: A vector in the Cartesian plane, showing the position of a point A with coordinates (2,3).

In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n-dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system. The endpoint of a vector can be identified with an ordered list of n real numbers (n-tuple). These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are typically called the scalar components (or scalar projections) of the vector on the axes of the coordinate system.

As an example in two dimensions, the vector from the origin O = (0,0) to the point A = (2,3) is simply written as a = (2,3). The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation OA is usually not deemed necessary and very rarely used. In three dimensional Euclidean space (or ℝ3), vectors are identified with triples of scalar components: a = (a2, a2, a3). also written a = (ax, ay, az These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows:

Another way to represent a vector in n-dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis of a Cartesian coordinate system, respectively, and they are sometimes referred to as versors of those axes. In terms of these, any vector a in JJ3can be expressed in the form: or where a1, a2, a3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z, while a1, a2, a3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often instead denoted i,j,k (or , in which the hat symbol ^ typically denotes unit vectors). xˆ, yˆ, zˆ In this case, the scalar and vector components are denoted ax, ay, az, and ax, ay, az. Thus,

a =ax +ay +az = axi+ayj+azk. The notation ei is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. Decomposition As explained above a vector is often described by a set of vector components that are mutually perpendicular and add up to form the given vector. Typically, these components are the projections of the vector on a set of reference axes (or basis vectors). The vector is said to be decomposed or resolved with respect to that set.

Figure: Illustration of tangential and normal components of a vector to a surface.

However, the decomposition of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian versors such as xˆ, yˆ, zˆ as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of the versors of a Cylindrical coordinate system ( ρˆ ,φˆ, zˆ ) or Spherical coordinate system (rˆ,θˆ ,φˆ ). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. The choice of a coordinate system doesn’t affect the properties of a vector or its behaviour under transformations. A vector can be also decomposed with respect to “non-fixed” axes which change their orientation as a function of time or space. For example, a vector in three dimensional space can be decomposed with respect to two axes, respectively normal, and tangent to a surface. Moreover, the radial and tangential components of a vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame).

Basic Properties The following section uses the Cartesian coordinate system with basis vectors e 1 = (1,0,0), e 2 = (0,1,0), e 3 = (0,0,1) and assumes that all vectors have the origin as a common base point. A vector a will be written as a = a1e 1+a2e 2 +a3e 3. Equality

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors a = a1e 1 +a2e 2 +a3e 3 and b = b1e 1 +b2e 2 +b3e 3 are equal if a1=b1, a2=b2, a3=b3.

Addition and subtraction Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is a + b = (a1 + b1)e 1 + (a2 + b2)e 2 + (a3 + b3)e 3. The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below:

This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, it will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is: a - b = (a1 - b1)e 1 + (a2 - b2)e 2 + (a3 - b3)e 3. Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the end points of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a “ b, as illustrated below:

Scalar Multiplication A vector may also be multiplied, or re-scaled, by a real number r. In the context of

conventional vector algebra, these real numbers are often called scalars (from scale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is called scalar multiplication. The resulting vector is ra = (ra1)e 1 + (ra2)e 2 + (ra3)e 3.

Figure: Scalar multiplication of a vector by a factor of 3 stretches the vector out.

Figure: The scalar multiplications 2a and -a of a vector a

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized (at least in the case when r is an integer) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector. If r is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = -1 and r = 2) are given below: Scalar multiplication is distributive over vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. Length The length or magnitude or norm of the vector a is denoted by ||a|| or, less commonly, |a|, which is not to be confused with the absolute value (a scalar “norm”). The length of the vector a can be computed with the Euclidean norm which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. This happens to be equal to the square root of the dot product, discussed below, of the vector with itself:

Null Vector In linear algebra, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero. It is usually written with an arrow head above or below it : 0 or 0 or simply 0. A zero vector has arbitrary direction, but is orthogonal (i.e. perpendicular, normal) to all other vectors with the same number of components. In vector spaces with an inner product for which the requirement of positive-definiteness has been dropped, a vector that has zero length is referred to as a null vector. The term zero vector is then still reserved for the additive identity of the vector spaces.

Linear Algebra For a general vector space, the zero vector is the uniquely determined vector that is the identity element for vector addition. The zero vector is unique; if a and b are zero vectors, then a = a + b = b. The zero vector is a special case of the zero tensor. It is the result of scalar multiplication by the scalar 0 (here meaning the additive identity of the underlying field, not necessarily the real number 0). The preimage of the zero vector under a linear transformation f is called kernel or null space. A zero space is a linear space whose only element is a zero vector. The zero vector is, by itself, linearly dependent, and so any set of vectors which includes it is also linearly dependent. In a normed vector space there is only one vector of norm equal to 0. This is just the zero vector. In vector algebra its coordinates are ( 0,0 ) and its unit vector is n. Seminormed Vector Spaces In a seminormed vector space there might be more than one vector of norm equal to 0. These vectors are often called null vectors. Examples The light-like vectors of Minkowski space are null vectors. In general, the coordinate representation of a null vector in Minkowski space contains non-zero values. In the Verma module of a Lie algebra there are null vectors.

Dot Product In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and then summing those products. The name “dot product” is derived from the centred dot “.” that is often used to designate this operation; the alternative name “scalar product” emphasizes the scalar (rather than vector) nature of the result. When two Euclidean vectors are expressed in terms of coordinate vectors on an orthonormal basis, the inner product of the former is equal to the dot product of the latter. For more general vector spaces, while both the inner and the dot product can be defined in different contexts (for instance with complex numbers as scalars) their definitions in these contexts may not coincide. In three dimensional space, the dot product contrasts with the cross product, which produces a vector as result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions. Definition The dot product of two vectors a = [a1, a2, ..., an ] and b = [b1, b2, ..., b n ] is defined as:

where Σ denotes summation notation and n is the dimension of the vector space.

In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. Similarly, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf For example, the dot product of two three-dimensional vectors [1, 3, -5] and [4, -2, -1] is [1,3, -5]·[4, -2,-1] = (1)(4) + (3)(-2) + (-5)(-1) = 4-6 + 5 = 3. Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix. The operation of extracting the coefficient of such a matrix can be written as taking its determinant or its trace (which is the same thing for 1 × 1 matrices); since in general tr(AB) = tr(BA) whenever AB or equivalently BA is a square matrix, one may write ab = det(aTb) = det(bTa) = tr(aTb) = tr(bTa) = tr(abT) = tr(baT) More generally the coefficient (ij) of a product of matrices is the dot product of the transpose of row i of the first matrix and column j of the second matrix.

Geometric Interpretation Since

Figure: θis the scalar projection of Aonto B.

In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector a , the dot product a·a is the square of the length (magnitude) of a, denoted by ||a||: If b is another such vector, and θ is the angle between them: This formula can be rearranged to determine the size of the angle between two nonzero vectors:

The Cauchy-Schwarz inequality guarantees that the argument of is valid. One can also first convert the vectors to unit vectors by dividing by their magnitude:

then the angle θ ισ given by θ = arccos(aˆ · b ˆ).

The terminal points of both unit vectors lie on the unit circle. The unit circle is where the trigonometric values for the six trig functions are found. After substitution, the first vector component is cosine and the second vector component is sine, i.e. (cos x, sin x) for some angle x. The dot product of the two unit vectors then takes (cos x, sin x) and (cos y, sin y) for angles and x and y returns cosxcos y + sinxsin y = cos(x-y) where x - y = θ. As the cosine of 90° is zero, the dot product of two orthogonal vectors is always zero. Moreover, two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length. This property provides a simple method to test the condition of orthogonality. Sometimes these properties are also used for “defining” the dot product, especially in 2 and 3 dimensions; this definition is equivalent to the above one. For higher dimensions the formula can be used to define the concept of angle. The geometric properties rely on the basis being orthonormal, i.e. composed of pairwise perpendicular vectors with unit length. Scalar Projection If both a and b have length one (i.e., they are unit vectors), their dot product simply gives the cosine of the angle between them. If only b is a unit vector, then the dot product a·b gives |a| cos θ, i.e., the magnitude of the projection of in the direction of b, with a minus sign if the direction is opposite. This is called the scalar projection of a onto b, or scalar component of in the direction of . This property of the dot product has several useful applications. If neither nor is a unit vector, then the magnitude of the projection of a in the direction of b is a , as the unit vector a in the direction of b is

.

Rotation When an orthonormal basis that the vector is represented in terms of is rotated, ‘s matrix in the new basis is obtained through multiplying by a rotation matrix . This matrix multiplication is just a compact representation of a sequence of dot products. For instance, let •

B1 = {x, y, z} and B2 = {B2 = {u, v, w} be two different orthonormal bases of the same space j|, with B2 obtained by just rotating B1,

• a = (a , a , a) represent vector a in terms of B, • a2 = (au, av, a w ) represent the same vector in terms of the rotated basis B2, • u1, v1, w1, be the rotated basis vectors u, v, w represented in terms of B Then the rotation from B, to B2 is performed as follows:

Notice that the rotation matrix R is assembled by using the rotated basis vectors u1, v1, w1 as its rows, and these vectors are unit vectors. By definition, Ra1 consists of a sequence of dot products between each of the three rows of R and vector a1. Each of these dot products determines a scalar component of a in the direction of a rotated basis vector. If a1 is a row vector, rather than a column

vector, then R must contain the rotated basis vectors in its columns, and must post-multiply a1:

Physics In physics, vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Example: • Mechanical work is the dot product of force and displacement vectors. • Magnetic flux is the dot product of the magnetic field and the area vectors. Properties The following properties hold if a, b, and c are real vectors and r is a scalar. The dot product is commutative: a·b = b·a. The dot product is distributive over vector addition: a(b + c) = ab + ac. The dot product is bilinear: a(rb + c) = r(a·b) + (a· c). When multiplied by a scalar value, dot product satisfies: (c1a)·(c2b) = (c1c2 )(a·b) (these last two properties follow from the first two). Two non-zero vectors a and b are orthogonal if and only if a • b = 0. Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law: If a • b = a • c and a ≠ 0, then we can write: a • (b - c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b - c), which still allows (b - c) ≠ 0, and therefore b ≠ c. Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions: • The new basis is again orthonormal (i.e., it is orthonormal expressed in the old one). • The new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis). If a and b are functions, then the derivative of a•b is a’ • b + a•b’

Triple Product Expansion This is a very useful identity (also known as Lagrange’s formula) involving the dot- and cross-products. It is written as a × (b × c) = b(ac) - c(ab) which is easier to remember as “BAC minus CAB”, keeping in mind which vectors are dotted together. This formula is commonly used to simplify vector calculations in physics. Proof of the Geometric Interpretation Consider the element of Rn Repeated application of the Pythagorean theorem yields for its length | v | | v|2=v12+v22 + ... + vn2. But this is the same as v·v = v2 + v2 +... + v2, so we conclude that taking the dot product of a vector v with itself yields the squared length of the vector. Lemma v.v =| v |2. Now consider two vectors a and b extending from the origin, separated by an angle θ. A third vector c may be defined as cdef = a-b. creating a triangle with sides a, b, and c. According to the law of cosines, we have |c|2 = |a|2 +|b|2 -2|a||b|cosθ. Substituting dot products for the squared lengths according to Lemma 1, we get c · c = aa + bb-2| a|| b | cosθ. But as c = a - b, we also have cc = (a-b)·(a-b) , which, according to the distributive law, expands to cc = aa + bb - 2(ab). Merging the two c • c equations, (1) and (2), we obtain aa + bb-2(ab) = aa + bb-2| a|| b | cosθ. Subtracting a • a + b • b from both sides and dividing by -2 leaves ab =| a || b | cosθ.

Generalization Real Vector Spaces The inner product generalizes the dot product to abstract vector spaces over the real numbers and is usually denoted by 〈a,b〉. More precisely, if V is a vector space over R, the inner product is a function V × V → ℝ. Owing to the geometric interpretation of the dot product, the norm ||a|| of a vector a in such an inner product space is defined as such that it generalizes length, and the angle 8 between two vectors a and b by

In particular, two vectors are considered orthogonal if their inner product is zero 〈a,b〉 = 0. Complex Vectors For vectors with complex entries, using the given definition of the dot product would lead to quite different geometric properties. For instance the dot product of a vector with itself can be an arbitrary complex number, and can be zero without the vector being the zero vector; this in turn would have severe consequences for notions like length and angle. Many geometric properties can be salvaged, at the cost of giving up the symmetric and bilinear properties of the scalar product, by alternatively defining where bi is the complex conjugate of bi . Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However this scalar product is not linear in b (but rather conjugate linear), and the scalar product is not symmetric either, since The angle between two complex vectors is then given by

This type of scalar product is nevertheless quite useful, and leads to the notions of Hermitian form and of general inner product spaces. The Frobenius inner product generalizes the dot product to matrices. It is defined as the sum of the products of the corresponding components of two matrices having the same size. Generalization to Tensors The dot product between a tensor of order n and a tensor of order m is a tensor of order n+m-2. The dot product is calculated by multiplying and summing across a single index in both tensors. If A and B are two tensors with element representation Ai k ’. “and Bmpn’i the elements of the dot product A·B are given by

This definition naturally reduces to the standard vector dot product when applied to vectors, and matrix multiplication when applied to matrices. Occasionally, a double dot product is used to represent multiplying and summing across two indices. The double dot product between two 2nd order tensors is a scalar quantity.

Cross Product In mathematics, the cross product, vector product, or Gibbs’ vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them. It has many applications in mathematics, engineering and physics.

Figure: The cross-product in respect to a right-handed coordinate system.

If either of the vectors being multiplied is zero or the vectors are parallel then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative, distributive over addition and satisfies the Jacobi identity. The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on the choice of orientation or “handedness”. The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n - 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. Definition The cross product of two vectors a and b is denoted by a × b. In physics, sometimes the notation a ^ b is used, though this is avoided in mathematics to avoid confusion with the exterior product.

Figure: Finding the direction of the cross product by the right-hand rule

The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span. The cross product is defined by the formula where θ is the measure of the smaller angle between a and b (0° ≤ 0 ≤ 180°), |a| and |b| are the magnitudes of vectors a and b, and n is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule as illustrated. If the vectors a and b are parallel (i.e., the angle #between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0. The direction of the vector n is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i.e., b × a = -(a × b). By pointing the forefinger toward b first, and then pointing the middle finger toward a, the thumb will be forced in the opposite direction, reversing the sign of the product vector. Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a lefthanded coordinate system is used, the direction of the vector n is given by the left-hand rule and points in the opposite direction. This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of n. The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. Names The cross product is also called vector product or Gibbs’ vector product. The name Gibbs’ vector product is after Josiah Willard Gibbs, who around 1881 introduced both the dot product and the cross product, using a dot (a · b) and a cross (a × b) to denote them.

Figure: Acco rd in g to S arru s’ rule, th e determin an t of a 3 ×3 matrix involves multiplications between matrix elements identified by crossed diagonals.

To emphasize the fact that the result of a dot product is a scalar, while the result of a

cross product is a vector, Gibbs also introduced the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation (a × b) and the name cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a · b involves multiplications between corresponding components of a and b. As explained below, the cross product can be defined as the determinant of a special 3×3 matrix. According to Sarrus’ rule, this involves multiplications between matrix elements identified by crossed diagonals.

Computing the Cross Product Coordinate Notation The standard basis vectors i, j, and k satisfy the following equalities: i×j = k, j×k = i, k × i = j, which imply, by the anticommutativity of the cross product, that j × i = -k, k × j = -i, i × k = -j

Figure: Standard basis vectors (i, j, k, also denoted e1 , e2 , e3 ) and vector components of a (a x, a, a z, also denoted a , a2, a 3

The definition of the cross product also implies that i×i = j×j = k×k = 0 (the zero vector). These equalities, together with the distributivity and linearity of the cross product, are sufficient to determine the cross product of any two vectors a and b. Each vector can be defined as the sum of three orthogonal components parallel to the standard basis vectors: a = a1 + a2 + a3 = a1 i + a2 j + a3k b = b1 +b2 +b3 =b1i+b2j + b3k Their cross product a × b can be expanded using distributivity:

a × b = (a 1i + a2j + a3k) × (b1i + b2j + b3k) = a 1 b 1i × i + a1b2i × j + a1b3i × k + a2b1 j × i + a2b2j × j + a2b3j × k + a 3 b 1k × i + a3b2k × j + a3b3k × k This can be interpreted as the decomposition of a × b into the sum of nine simpler cross products involving vectors aligned with i, j, or k. Each one of these nine cross products operates on two vectors that are easy to handle as they are either parallel or orthogonal to each other. From this decomposition, by using the above mentioned equalities and collecting similar terms, we obtain: a × b = a1b1 0 + a1 b 2k + a1 b 3 (-j) + a2b1 (-k) + a2b2 0 + a2b3i + a3bj + a3b2(-i) + a3b30 = (a2b3 -a3b2)i + (a3 b 1 -a1 b 3)j + (a 1 b2 -a2 b 1)k. meaning that the three scalar components of the resulting vector c = c1i + cj + c3k = a × b are c 1=a2b3-a3b2 c2 = a3b1 - a1b3 c3 = a1b2 - a2b1 Using column vectors, we can represent the same result as follows:

Matrix Notation The definition of the cross product can also be represented by the determinant of a formal matrix:

This determinant can be computed using Sarrus’ rule or Cofactor expansion. Using Sarrus’ Rule, it expands to Using Cofactor expansion along the first row instead, it expands to

which gives the components of the resulting vector directly. Properties Geometric meaning: The magnitude of the cross product can be interpreted as the positive area of the parallelogram having a and b as sides: A = |a×b|=|a||b|sinθ. Indeed, one can also compute the volume V of a parallelepiped having a, b and c as sides by using a combination of a cross product and a dot product, called scalar triple product:

a(b × c) = b(c × a) = c(a × b).

Figure: The area of a parallelogram as a cross product

Figure: Three vectors defining a parallelepiped

Since the result of the scalar triple product may be negative, the volume of the parallelepiped is given by its absolute value. For instance, V=|a·(b×c)|. Because the magnitude of the cross product goes by the sine of the angle between its arguments, the cross product can be thought of as a measure of “perpendicularness” in the same way that the dot product is a measure of “parallelness”. Given two unit vectors, their cross product has a magnitude of 1 if the two are perpendicular and a magnitude of zero if the two are parallel. The opposite is true for the dot product of two unit vectors. Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors. The magnitude of the cross product of the two unit vectors yields the sine (which will always be positive). Algebraic Properties The cross product is anticommutative, a×b = -b×a, distributive over addition, a × (b + c) = (a × b) + (a × c), and compatible with scalar multiplication so that (ra) × b = a × (rb) = r(a × b). It is not associative, but satisfies the Jacobi identity: a × (b × c) + b × (c × a) + c × (a × b) = 0. Distributivity, linearity and Jacobi identity show that R3 together with vector addition and the cross product forms a Lie algebra, the Lie algebra of the real orthogonal group in 3 dimensions, SO(3).

The cross product does not obey the cancellation law: a × b = a × c with non-zero a does not imply that b = c. Instead if a × b = a × c:

If neither a nor b - c is zero then from the definition of the cross product the angle between them must be zero and they must be parallel. They are related by a scale factor, so one of b or c can be expressed in terms of the other, for example c = b + ta, for some scalar t. If a b = a c and a × b = a × c, for non-zero vector a, then b = c, as

So b - c is both parallel and perpendicular to the non-zero vector a, something that is only possible if b - c = 0 so they are identical. From the geometrical definition the cross product is invariant under rotations about the axis defined by a × b. More generally the cross product obeys the following identity under matrix transformations: (Ma) × (Mb) = (detM)M~T (a × b) where is a 3 by 3 matrix and M-T is the transpose of the inverse The cross product of two vectors in 3-D always lies in the null space of the matrix with the vectors as rows:

For the sum of two cross products, the following identity holds: Differentiation The product rule applies to the cross product in a similar manner:

This identity can be easily proved using the matrix multiplication representation. Triple Product Expansion The cross product is used in both forms of the triple product. The scalar triple product of three vectors is defined as a·(b×c), It is the signed volume of the parallelepiped with edges a, b and c and as such the vectors can be used in any order that’s an even permutation of the above ordering. The following therefore are equal: a·(b × c) = b·(c × a) = c·(a × b), The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following formula

a × (b × c) = b(ac) - c(ab). The mnemonic “BAC minus CAB” is used to remember the order of the vectors in the right hand member. This formula is used in physics to simplify vector calculations. A special case, regarding gradients and useful in vector calculus, is

where ∇2 is the vector Laplacian operator. Another identity relates the cross product to the scalar triple product: (a × b) × (a × c) = (a(b × c))a Alternative Formulation The cross product and the dot product are related by: The right-hand side is the Gram determinant of a and b, the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle θ between the two vectors, as: the above given relationship can be rewritten as follows: Invoking the Pythagorean trigonometric identity one obtains: which is the magnitude of the cross product expressed in terms of θ, equal to the area of the parallelogram defined by a and b. The combination of this requirement and the property that the cross product be orthogonal to its constituents a and b provides an alternative definition of the cross product. Lagrange’s Identity The relation: |a×b|2=|a|2|b|2-(a·b)2. can be compared with another relation involving the right-hand side, namely Lagrange’s identity expressed as:

where a and b may be n-dimensional vectors. In the case n=3, combining these two equations results in the expression for the magnitude of the cross product in terms of its components: The same result is found directly using the components of the cross-product found from:

In R3 Lagrange’s equation is a special case of the multiplicativity | vw | = | v | | w | of the norm in the quaternion algebra. It is a special case of another formula, also sometimes called Lagrange’s identity, which is the three dimensional case of the Binet-Cauchy identity: (a × b)·(c × d) = (a·c)(b·d) - (a·d)(b·c). If a = c and b = d this simplifies to the formula above. Alternative Ways to Compute the Cross Product Conversion to matrix multiplication: The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:

where superscript T refers to the transpose operation, and [a]× is defined by:

Also, if a is itself a cross product: a=c×d then This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors. This notation is also often much easier to work with, for example, in epipolar geometry. From the general properties of the cross product follows immediately that [a]×a = 0 and aT[a]× =0 and from fact that [a]× is skew-symmetric it follows that

bT[a]×b = 0. The above-mentioned triple product expansion (bac-cab rule) can be easily proven using this notation. The above definition of [a]× means that there is a one-to-one mapping between the set of 3×3 skew-symmetric matrices, also known as the Lie algebra of SO(3), and the operation of taking the cross product with some vector a. Index Notation for Tensors The cross product can alternatively be defined in terms of the Levi-Civita symbol, åijk which is useful in converting vector notation for tensor applications:

where the indices i, j,k correspond, as in the previous section, to orthogonal vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as in which repeated indices are summed from 1 to 3. Note that this representation is another form of the skew-symmetric representation of the cross product: In Classical Mechanics: representing the cross-product with the Levi-Civita symbol can cause mechanical-symmetries to be obvious when physical-systems are isotropic in space. (Quick example: consider a particle in a Hooke’s Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are “special” in any sense, so symmetries lie in the cross-product-represented angular-momentum which are made clear by the above mentioned Levi-Civita representation). Mnemonic The word “xyzzy” can be used to remember the definition of the cross product. If a = b×c where:

then:

The second and third equations can be obtained from the first by simply vertically rotating the subscripts, x → y → z → x. The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus’s scheme (those containing i), or to remember the xyzzy sequence. Since

the first diagonal in Sarrus’s scheme is just the main diagonal of the above-mentioned matrix, the first three letters of the word xyzzy can be very easily remembered. Cross Visualization Similarly to the mnemonic device above, a “cross” or × can be visualized between the two vectors in the equation. This may help you to remember the correct cross product formula. If a = b × c then:

If we want to obtain the formula for ax we simply drop the bx and cx from the formula, and take the next two components down

It should be noted that when doing this for ay the next two elements down should “wrap around” the matrix so that after the z component comes the × component. For clarity, when performing this operation for ay , the next two components should be z and × (in that order). While for az the next two components should be taken as × and y.

For ax then, if we visualize the cross operator as pointing from an element on the left to an element on the right, we can take the first element on the left and simply multiply by the element that the cross points to in the right hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our a formula: We can do this in the same way for ay and az to construct their associated formulas. Applications Computational Geometry: The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of polygon (clockwise or anticlockwise) about a point within the polygon (i.e. the centroid or mid-point) can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle. In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points p1 = (x1, y1), p2 (x2, y2). It corresponds to the direction of the cross product of the two coplanar vectors defined by the pairs of points p1, p2 and p1, p3, i.e., by the sign of the expression P = (x2, x1) (y3 – y1) – (y2 – y1) (x3 – x1). In the “right-handed” coordinate system, if the result is 0, the points are

collinear; if it is positive, the three points constitute a negative angle of rotation around from to , otherwise a positive angle. From another point of view, the sign of tells whether lies to the left or to the right of line p1,p2 . Mechanics Moment of a force applied at point B around point A is given as: MA =rAB ×FB . Other The cross product occurs in the formula for the vector operator curl. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product. The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints. Cross Product as an Exterior Product The cross product in relation to the exterior product. In red are the orthogonal unit vector, and the “parallel” unit bivector. The cross product can be viewed in terms of the exterior product. This view allows for a natural geometric interpretation of the cross product. In exterior algebra the exterior product (or wedge product) of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element.

Given two vectors a and b, one can view the bivector a A b as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge dual of the bivector a ^ b, mapping 2-vectors to vectors: a × b = *(a ^ b). This can be thought of as the oriented multidimensional element “perpendicular” to the bivector. Only in three dimensions is the result an oriented line element – a vector – whereas, for example, in 4 dimensions the Hodge dual of a bivector is two-dimensional – another oriented plane element. So, only in three dimensions is the cross product of a and b the vector dual to the bivector a ^ b: it is perpendicular to the bivector, with orientation dependent on the coordinate system’s handedness, and has the same magnitude relative to the unit normal vector as a ^ b has relative to the unit bivector; precisely the properties described above. Cross Product and Handedness

When measurable quantities involve cross products, the handedness of the coordinate systems used cannot be arbitrary. However, when physics laws are written as equations, it should be possible to make an arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two vectors, one must take into account that when the handedness of the coordinate system is not fixed a priori, the result is not a (true) vector but a pseudovector. Therefore, for consistency, the other side must also be a pseudovector. More generally, the result of a cross product may be either a vector or a pseudovector, depending on the type of its operands (vectors or pseudovectors). Namely, vectors and pseudovectors are interrelated in the following ways under application of the cross product: • vector × vector = pseudovector • pseudovector × pseudovector = pseudovector • vector × pseudovector = vector • pseudovector × vector = vector. So by the above relationships, the unit basis vectors i, j and k of an orthonormal, righthanded (Cartesian) coordinate frame must all be pseudovectors (if a basis of mixed vector types is disallowed, as it normally is) since i × j = k, j × k = i and k × i = j. Because the cross product may also be a (true) vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a (true) vector and the other one is a pseudovector (e.g., the cross product of two vectors). For instance, a vector triple product involving three (true) vectors is a (true) vector. A handedness-free approach is possible using exterior algebra. Generalizations There are several ways to generalize the cross product to the higher dimensions. Lie Algebra The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory. For example, the Heisenberg algebra gives another Lie algebra structure on R3. In the basis {x, y, z}, the product is [x, y] = z[x, z] = [y, z] = 0. Quaternions The cross product can also be described in terms of quaternions, and this is why the letters i, j, k are a convention for the standard basis on R3. The unit vectors i, j, k correspond to “binary” (180 deg) rotations about their respective axes (Altmann, S. L., 1986, Ch. 12), said rotations being represented by “pure” quaternions (zero scalar part) with unit norms. For instance, the above given cross product relations among i, j, and k agree with the

multiplicative relations among the quaternions i, j, and k. In general, if a vector [a1, a2, a3] is represented as the quaternion a1i + a2j + a3k, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors. Alternatively and more straightforwardly, using the above identification of the ‘purely imaginary’ quaternions with R3, the cross product may be thought of as half of the commutator of two quaternions. Octonions A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of such cross products of two vectors in other dimensions is related to the result that the only normed division algebras are the ones with dimension 1, 2, 4, and 8; Hurwitz’s theorem. Wedge Product In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to map 2-vectors to vectors. The Hodge dual of the wedge product yields an (n-2)-vector, which is a natural generalization of the cross product in any number of dimensions. The wedge product and dot product can be combined (through summation) to form the geometric product. Multilinear Algebra In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index. In detail, the 3-dimensional volume form defines a product V×V×V→R, by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function V ×V → V*, (fixing any two inputs gives a function V → R by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism V → V*, and thus this yields a map V ×V → V, which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by “raising an index”. Translating the above algebra into geometry, the function “volume of the parallelepiped defined by (a, b, -) “ (where the first two vectors are fixed and the last is an input), which defines a function V → R , can be represented uniquely as the dot product with a vector: this vector is the cross product a × b. From this perspective, the cross product is defined by the scalar triple product Vol(a, b, c) = (a×b)·c. In the same way, in higher dimensions one may define generalized cross products by

raising indices of the n-dimensional volume form, which is a -tensor. The most direct generalizations of the cross product are to define either: • a (1,n-1) -tensor, which takes as input vectors, and gives as output 1 vector - an (n -1) -ary vector-valued product, or • a (n - 2,2) -tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank n-2 – a binary product with rank n-2 tensor values. One can also define (k, n-k) -tensors for other k. These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity. The -ary product can be described as follows: given vectors in define their generalized cross product as: • perpendicular to the hyperplane defined by the vi, • magnitude is the volume of the parallelotope defined by the vi, which can be computed as the Gram determinant of the • oriented so that v1,...,vnis positively oriented vi,. This is the unique multilinear, alternating product which evaluates to e1 x • • • x en1 = en, e2x---xen= e1, and so forth for cyclic permutations of indices. In coordinates, one can give a formula for this (n -1) -ary analogue of the cross product in Rn by:

This formula is identical in structure to the determinant formula for the normal cross product in R3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v1,...,vn-1, ∧ (v1,...,vn-1)) have a positive orientation with respect to (e1,...,en). If n is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is even, however, the distinction must be kept. This (n -1) -ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments. History In 1773, Joseph Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. In 1843 the Irish mathematical physicist Sir William Rowan Hamilton introduced the quaternion product, and with it the

terms “vector” and “scalar”. Given two quaternions [0, u] and [0, v], where u and v are vectors in R3, their quaternion product can be summarized as [-u·v, u×v]. James Clerk Maxwell used Hamilton’s quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education. In 1878 William Kingdon Clifford published his Elements of Dynamic which was an advanced text for its time. He defined the product of two vectors to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane. Oliver Heaviside in England and Josiah Willard Gibbs, a professor at Yale University in Connecticut, also felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced—to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell’s original 20 to the four commonly seen today. Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. William Kingdon Clifford combined the algebras of Hamilton and Grassmann to produce Clifford algebra, where in the case of three-dimensional vectors the bivector produced from two vectors dualizes to a vector, thus reproducing the cross product. The cross notation and the name “cross product” began with Gibbs. Originally they appeared in privately published notes for his students in 1881 as Elements of Vector Analysis. The utility for mechanics was noted by Aleksandr Kotelnikov. Gibbs’s notation and the name “cross product” later reached a wide audience through Vector Analysis, a textbook by Edwin Bidwell Wilson, a former student. Wilson rearranged material from Gibbs’s lectures, together with material from publications by Heaviside, Fopps, and Hamilton. He divided vector analysis into three parts: First, that which concerns addition and the scalar and vector products of vectors. Second, that which concerns the differential and integral calculus in its relations to scalar and vector functions. Third, that which contains the theory of the linear vector function. Two main kinds of vector multiplications were defined, and they were called as follows: • The direct, scalar, or dot product of two vectors • The skew, vector, or cross product of two vectors Several kinds of triple products and products of more than three vectors were also examined. The above mentioned triple product expansion was also included.

Level Set

In mathematics, a level set of a real-valued function f of n variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. When n = 3, a level set is called a level surface, and for higher values of n the level set is a level hypersurface. A set of the form is called a sublevel set of f (or, alternatively, a lower level set or trench of f). is called a superlevel set of f. A level set is a special case of a fibre. Properties • The gradient of f at a point is perpendicular to the level set of f at that point. • The sublevel sets of a convex function are convex (the converse is however not generally true).

Divergence In vector calculus, divergence is a vector operator that measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. Definition of Divergence In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. It is a local measure of its “outgoingness”—the extent to which there is more exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point then there must be a source or sink at that position. (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink and so on.) More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

where | V | is the volume of V, S(V) is the boundary of V, and the integral is a surface integral

with n being the outward unit normal to that surface. The result, div F, is a function of p. From this definition it also becomes explicitly visible that div F can be seen as the source density of the flux of F. In light of the physical interpretation, a vector field with constant zero divergence is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface. The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem. Application in Cartesian Coordinates Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors. The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests. The common notation for the divergence A F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of A(del), apply them to the components of F, and sum the results. Because applying an operator is different from multiplying the components, this is considered an abuse of notation. The divergence of a continuously differentiable second-rank tensor field tensor field:

is a first-rank

Cylindrical Coordinates For a vector expressed in cylindrical coordinates as where ea is the unit vector in direction a, the divergence is

Spherical Coordinates In spherical coordinates, with 8 xr|e angle with the z axis and ϕ rotation around the z axis, the divergence reads

Chapter 2: Coordinate Expressions Two Dimensions The Laplace operator in two dimensions is given by

where x and y are the standard Cartesian coordinates of the xy-plane. In polar coordinates,

Three Dimensions In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems. In Cartesian coordinates,

In cylindrical coordinates,

In spherical coordinates:

(here cp represents the azimuthal angle and 8 the zenith angle or co-latitude). In general curvilinear coordinates (ξ1,ξ2,ξ3):

where summation over the repeated indices is implied.

N Dimensions In spherical coordinates in N dimensions, with the parametrization x = rθ ∞ RN with r representing a positive real radius and 6 an element of the unit sphere SN -1,

where ΔSN-1 is the Laplace–Beltrami operator on the (N-1)-sphere, known as the spherical Laplacian. The two radial terms can be equivalently rewritten as

As a consequence, the spherical Laplacian of a function defined on SN-1 ⊂ RN can be computed as the ordinary Laplacian of the function extended to RN \{0} so that it is constant along rays, i.e., homogeneous of degree zero. Spectral Theory The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with -Δf = λf. If Ψ is a bounded domain in Rn then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L2(Ψ). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Q. is the n-sphere, the eigenfunctions of the Laplacian are the well-known spherical harmonics.

Laplace-Beltrami Operator In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace-Beltrami operator, after Laplace and Eugenio Beltrami. Like the Laplacian, the Laplace-Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is called the Laplace-de Rham operator (named after Georges de Rham). The Laplace-Beltrami operator, like the Laplacian, is the divergence of the gradient: Δf = div grad f. An explicit formula in local coordinates is possible. Suppose first that M is an oriented Riemannian manifold. The orientation allows one to specify a definite volume form on M, given in an oriented coordinate system xi by where the dxi are the 1-forms forming the dual basis to the basis vectors

and is the wedge product. Here |g| := |det(gi)| is the absolute value of the determinant of the metric tensor gij . The divergence div X of a vector field X on the manifold is then defined as the scalar function with the property (divX) voln := LXvoln where LX is the Lie derivative along the vector field X. In local coordinates, one obtains

where the Einstein notation is implied, so that the repeated index i is summed over. The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product 〈•, •〉 on the manifold, as 〈gradf(x),vx〉 = df(x)(vx) for all vectors vx anchored at point x in the tangent space TxM of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx . In local coordinates, one has where gij are the components of the inverse of the metric tensor, so that gij gjk = δik with δik the Kronecker delta. Combining the definitions of the gradient and divergence, the formula for the Laplace– Beltrami operator A applied to a scalar function ƒ is, in local coordinates

If M is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a volume element (a density rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace–Beltrami operator itself does not depend on this additional structure. Formal Self-Adjointness The exterior derivative d and “div are formal adjoints, in the sense that for ƒ a compactly supported function where the last equality is an application of Stokes’ theorem. Dualizing gives for all compactly supported functions ƒ and h. Conversely, (2) characterizes Δ completely, in the sense that it is the only operator with this property. As a consequence, the LaplaceBeltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions ƒ and h,

Because the Laplace–Beltrami operator, as defined in this manner, is negative rather than positive, often it is defined with the opposite sign. Tensor Laplacian The Laplace-Beltrami operator can be written using the trace of the iterated covariant derivative associated to the Levi-Civita connection. From this perspective, let X i be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the Hessian of a function f is the symmetric 2-tensor whose components are given by This is easily seen to transform tensorially, since it is linear in each of the arguments Xi, Xj. The Laplace-Beltrami operator is then the trace of the Hessian with respect to the metric: In abstract indices, the operator is often written provided it is understood implicitly that this trace is in fact the trace of the Hessian tensor. Because the covariant derivative extends canonically to arbitrary tensors, the LaplaceBeltrami operator defined on a tensor T by is well-defined. Laplace-de Rham Operator More generally, one can define a Laplacian differential operator on sections of the bundle of differential forms on a pseudo-Riemannian manifold. On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace-de Rham operator is defined by Δ = dδ + δd = (d + δ)2, where d is the exterior derivative or differential and δ is the codifferential, acting as kn+n+1 (-1) ∗ d ∗ on k-forms where ∗ is the Hodge star. When computing Δ ƒ for a scalar function ƒ, we have δ ƒ = 0, so that Δ f = δdf. Up to an overall sign, The Laplace-de Rham operator is equivalent to the previous definition of the Laplace-Beltrami operator when acting on a scalar function. On functions, the Laplace-de Rham operator is actually the negative of the LaplaceBeltrami operator, as the conventional normalization of the codifferential assures that the Laplace-de Rham operator is (formally) positive definite, whereas the Laplace-Beltrami operator is typically negative. The sign is a pure convention, however, and both are common in the literature. The Laplace–de Rham operator differs more significantly from the tensor Laplacian restricted to act on skew-symmetric tensors. Apart from the incidental sign, the two operators differ by a Weitzenböck identity that

explicitly involves the Ricci curvature tensor. Examples Many examples of the Laplace-Beltrami operator can be worked out explicitly. Euclidean Space In the usual (orthonormal) Cartesian coordinates xi on Euclidean space, the metric is reduced to the Kronecker delta, and one therefore has | g |= 1. Consequently, in this case

which is the ordinary Laplacian. In curvilinear coordinates, such as spherical or cylindrical coordinates, one obtains alternative expressions. Similarly, the Laplace–Beltrami operator corresponding to the Minkowski metric with signature (“+++) is the D’Alembertian. Spherical Laplacian The spherical Laplacian is the Laplace-Beltrami operator on the (n - 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded into Rn as the unit sphere centred at the origin. Then for a function ƒ on Sn-1, the spherical Laplacian is defined by where ƒ(x/| x |) is the degree zero homogeneous extension of the function ƒ to Rn - {0}, and A is the Laplacian of the ambient Euclidean space. Concretely, this is implied by the wellknown formula for the Euclidean Laplacian in spherical polar coordinates:

More generally, one can formulate a similar trick using the normal bundle to define the Laplace-Beltrami operator of any Riemannian manifold isometrically embedded as a hypersurface of Euclidean space. One can also give an intrinsic description of the Laplace-Beltrami operator on the sphere in a normal coordinate system. Let (t, d) be spherical coordinates on the sphere with respect to a particular point p of the sphere (the “north pole”), that is geodesic polar coordinates with respect to p. Here t represents the latitude measurement along a unit speed geodesic from p, and d a parameter representing the choice of direction of the geodesic in Sn-1. Then the spherical Laplacian has the form:

where Δζ is the Laplace-Beltrami operator on the ordinary unit (n - 2)-sphere. Hyperbolic Space A similar technique works in hyperbolic space. Here the hyperbolic space Hn -1 can be embedded into the n dimensional Minkowski space, a real vector space equipped with the quadratic form

Then Hn is the subset of the future null cone in Minkowski space given by Hn = {x|q(x) = 1, x1 > 1}. Then Here f(x / q(x)1/2) is the degree zero homogeneous extension of f to the interior of the future null cone and a is the wave operator

The operator can also be written in polar coordinates. Let (t, ∂) be spherical coordinates on the sphere with respect to a particular point p of Hn-1 (say, the centre of the Poincaré disc). Here t represents the hyperbolic distance from p and #x2202; a parameter representing the choice of direction of the geodesic in Sn -1. Then the spherical Laplacian has the form:

where Δζ is the Laplace-Beltrami operator on the ordinary unit (n - 2)-sphere. D’Alembertian The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic. In the Minkowski space the Laplace–Beltrami operator becomes the d’Alembert operator or d’Alembertian:

It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time independent functions. Note that the overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high energy particle physics. The D’Alembert operator is also known as the wave operator, because it is the differential operator appearing in the wave equations and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case. The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in metres while the y direction were measured in centimetres. Indeed, theoretical physicists usually work in units such that c=1 in order to simplify the equation.

Gradient Theorem The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field (any conservative vector field can be expressed as the gradient of a scalar field) can be evaluated by evaluating the original scalar field at the endpoints of the curve:

It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. The gradient theorem implies that line integrals through irrotational vector fields are path independent. In physics this theorem is one of the ways of defining a “conservative” force. By placing ϕ as potential, Δϕ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. Proof If is a differentiable function from some subset U ⊆ ℝn to R, and if r is a continuous function from some closed interval [a,b]to U, then by the multivariate chain rule, the equality holds for all t ∞ [a, b], where the denotes the usual inner product. Therefore, r if parametrizes the (continuous) curve γ ⊂ U with endpoints p, q (oriented in the direction from P to q) for t ∞ [a, b], then

where we have used the definition of the line integral in the first equality, and the fundamental theorem of calculus in the third equality. Examples Suppose γ ⊂ ℝ2 is the circular arc oriented counterclockwise from (5,0) to (-4,3). Using the definition of a line integral,

Notice all of the painstaking computations involved in directly calculating the integral. Instead, since the function f(x, y) = xy is differentiable on all of ℝ2, we can simply use the gradient theorem to say

Notice that either way gives the same answer, but using the latter method, most of the work is already done in the proof of the gradient theorem. Generalizations

In the language of differential forms and exterior derivatives, the gradient theorem states that for any 0-form ϕ defined on some differentiable curve γ ⊂ ℝn (here the integral of (betnep the boundary of the curve y is understood to be the difference in value of ϕ βɛτωɛɛv the endpoints of the curve). Notice the striking similarity between this statement and the generalized version of Stokes’ theorem, which says that the integral of any compactly supported differential form mover the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative do over the whole of QΩ, i.e. This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension.

Green’s Theorem In mathematics, Green’s theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes’ theorem, and is named after British mathematician George Green. Theorem Let C be a positively oriented, piecewise smooth, simple closed curve in the plane 2, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol. In physics, Green’s theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green’s theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. Proof When D is a Simple Region If D is a simple region with its boundary consisting of the curves C1, C2, C3, C4, Green’s theorem can be demonstrated. The following is a proof of the theorem for the simplified area D, a type I region where C2 and C4 are vertical lines. A similar proof exists for when D is a type II region where C1 and C3 are straight lines. The general case can be deduced from this special case by approximating the domain D by a union of simple domains.

If it can be shown that

are true, then Green’s theorem is proven in the first case. Define the type I region D as pictured on the right by where g1 and g2 are continuous functions on [a, b]. Compute the double integral in (1):

Now compute the line integral in (1). C can be rewritten as the union of four curves: C1, C2, C3, C4. With C1, use the parametric equations: x = x, y = g1(x), a ≤ x ≤ b. Then With C3, use the parametric equations: x = x, y = g 2(x), a ≤ x ≤ b. Then The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively (counterclockwise). On C2 and C4, x remains constant, meaning Therefore,

Combining (3) with (4), we get (1). Similar computations give (2). Relationship to the Stokes Theorem Green’s theorem is a special case of the Kelvin-Stokes theorem, when applied to a region in the xy-plane: We can augment the two-dimensional field into a three-dimensional field with a z component that is always 0. Write F for the vector-valued function F =(L,M,0). Start with the left side of Green’s theorem:

Then by Kelvin-Stokes Theorem: The surface S is just the region in the plane D, with the unit normals n ˆ pointing up (in the positive z direction) to match the “positive orientation” definitions for both theorems. The expression inside the integral becomes

Thus we get the right side of Green’s theorem

Relationship to the Divergence Theorem Considering only two-dimensional vector fields, Green’s theorem is equivalent to the following two-dimensional version of the divergence theorem: where n̂ is the outward-pointing unit normal vector on the boundary. To see this, consider the unit normal n in the right side of the equation. Since in Green’s theorem dr = (dx, dy) is a vector pointing tangential along the curve, and the curve C is the positively-oriented (i.e. counterclockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right, which would be (dy, -dx). The length of this vector is So Now let the components of F = (P, Q). Then the right hand side becomes which by Green’s theorem becomes

The converse can also easily shown to be true. Area Calculation Green’s theorem can be used to compute area by line integral. The area of D is given by: Provided we choose L and M such that:

Then the area is given by: Possible formulas for the area of D include:

Divergence Theorem In vector calculus, the divergence theorem, also known as Ostrogradsky’s theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. The theorem is a special case of the more general Stokes’ theorem.

Intuition If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field’s divergence at a given point describes the strength of the source or sink there. So, integrating the field’s divergence over the interior of the region should equal the integral of the vector field over the region’s boundary. The divergence theorem says that this is true. The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume’s boundary. Mathematical Statement Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighbourhood of V, then we have

Figure: A region V bounded by the surface S=3V with the surface

Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighbourhood of V, then we have

Figure: The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left. It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outwardpointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for n dS.) By the symbol within the two integrals it is stressed once more that ∂V is a closed surface. In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary ∂V. Corollaries By applying the divergence theorem in various contexts, other useful identities can be derived (cf vector identities). •

Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

A special case of this is F = Δf , in which case the theorem is the basis for Green’s identities. • Applying the divergence theorem to the cross-product of two vector fields F Δ G, the result is • Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven: • Applying the divergence theorem to the cross-product of a vector field F and a nonzero constant vector, the following theorem can be proven:

Example

Suppose we wish to evaluate where S is the unit sphere defined by and F is the vector field The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem:

where W is the unit ball (i.e., the interior of the unit sphere, x2 + y2 + z2 ≤ 1). Since the function is positive in one hemisphere of W and negative in the other, in an equal and opposite way, its total integral over W is zero. The same is true for z: Therefore,

because the unit ball W has volume

.

Applications Differential form and Integral form of Physical Laws: As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are Gauss’s law (in electrostatics), Gauss’s law for magnetism, and Gauss’s law for gravity.

Chapter 3: Continuity Equation A continuity equation in physics is an equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. Continuity equations are the (stronger) local form of conservation laws. All the examples of continuity equations below express the same idea, which is roughly that: the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place. Any continuity equation can be expressed in an “integral form” (in terms of a flux integral), which applies to any finite region, or in a “differential form” (in terms of the divergence operator) which applies at a point. Continuity equations underlie more specific transport equations such as the convection– diffusion equation, Boltzmann transport equation, and Navier-Stokes equations.

General Equation Preliminary description: As stated above, the idea behind the continuity equation is the flow of some property, such as mass, energy, electric charge, momentum, and even probability, through surfaces from one region of space to another. The surfaces, in general, may either be open or closed, real or imaginary, and have an arbitrary shape, but are fixed for the calculation (i.e. not time-varying, which is appropriate since this complicates the maths for no advantage).

Figure: Illustration of how flux f passes through open surfaces S (vector S), flat or curved.

Figure: Illustration of how flux f passes through closed surfaces S 1 and S 2 . The surface area elements shown are dS 1 and dS 2 , and the flux is integrated over the whole surface. Yellow dots are sources, red dots are sinks, the blue lines are the flux lines of q.

Let this property be represented by just one scalar variable, q, and let the volume density of this property (the amount of q per unit volume V) be ϕ, and the all surfaces be denoted by S. Mathematically, ϕ is a ratio of two infinitesimal quantities:

which has the dimension [quantity][L]-3 (where L is length). There are different ways to conceive the continuity equation: 1. Either the flow of particles carrying the quantity q, described by a velocity field v, which is also equivalent to a flux f of q (a vector function describing the flow per unit area per unit time of q), or 2. In the cases where a velocity field is not useful or applicable, the flux f of the quantity q only (no association with velocity). In each of these cases, the transfer of q occurs as it passes through two surfaces, the first S1 and the second S2.

Figure: Illustration of q, ϕ, and f, and the effective flux due to carriers of q. ϕ is the amount of q per unit volume (in the box), f represents the flux (blue flux lines) and q is carried by the particles (yellow).

The flux f should represent some flow or transport, which has dimensions [quantity] [T]-1[L]-2. In cases where particles/carriers of quantity q are moving with velocity v, such as particles of mass in a fluid or charge carriers in a conductor, f can be related to v by: f = ϕv. This relation is only true in situations where there are particles moving and carrying q it can’t always be applied. To illustrate this: if f is electric current density (electric current per unit area) and ϕ is the charge density (charge per unit volume), then the velocity of the charge carriers is v. However - if f is heat flux density (heat energy per unit time per unit area), then even if we let ϕ be the heat energy density (heat energy per unit volume) it does not imply the “velocity of heat” is v (this makes no sense, and is not practically applicable). In the latter case only f (with ϕ) may be used in the continuity equation.

Elementary Vector Form Consider the case when the surfaces are flat and planar cross-sections. For the case where a velocity field can be applied, dimensional analysis leads to this form of the continuity equation: where • The left hand side is the initial amount of q flowing per unit time through surface S1, the right hand side is the final amount through surface S2, • S1 and S2 are the vector areas for the surfaces S1 and S2 respectively. Notice the dot products v1?S1,v2?S2 are volumetric flow rates of q. The dimen sion o f each side of the equat ion is [quan tity][ L]-3•[L][T]-1•[L]2 = [quantity][T]-1. For the more general cases, independent of whether a velocity field can be used or not, the continuity equation becomes: f1?S1=f2?S2 This has exactly the same dimensions as the previous version. The relation between f and v allows us to pass back to the velocity version from this flux equation, but not always the other way round (as explained above - velocity fields are not always applicable). These results can be generalized further to curved surfaces by reducing the vector surfaces into infinitely many differential surface elements (that is S → dS), then integrating over the surface: more generally still: in which •

denotes a surface integral over the surfaceS,

• n̂ is the outward-pointing unit normal to the surface SN.B: the scalar area S and vector area S are related by dS = n̂dS . Either notations may be used interchangeably. Differential Form The differential form for a general continuity equation is (using the same q, φ and f as above): where • • • •

Δ is divergence, t is time, σ is the generation of q per unit volume per unit time.Terms that generate (Σ > 0) or remove (σ < 0) q arereferred to as a “sources” and “sinks” respectively.

This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier-Stokes equations. This equation also generalizes the advection equation. Other equations in physics, such as Gauss’s law of the electric field and Gauss’s law for gravity, have a similar mathematical form to the continuity equation, but are not usually called by the term “continuity equation”, because f in those cases does not represent the flow of a real physical quantity. In the case that q is a conserved quantity that cannot be created or destroyed (such as energy), this translates to σ = 0, and the continuity equation is:

Integral Form By the divergence theorem, the continuity equation can be rewritten in an equivalent way, called the “integral form”:

where • S is a surface as described above - except this time it has to be a closed surface that encloses a volume V, denotes a surface integral over a closed surface, • denotes a volume integral over V. • is the total amount of φ in the volume V; • is the total generation (negative in the case of removal) per unit time by the • sources and sinks in the volume V,

Figure: In the integral form of the continuity equation, S is any imaginary closed surface that fully encloses a volume V, like any of the surfaces on the left. S can not be a surface with boundaries that do not enclose a volume, like those on the right. (Surfaces are blue, boundaries are red.)

In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a “source” where σ > 0), and decreases when someone in the building dies (a “sink” where σ < 0).

Derivation and Equivalence The differential form can be derived from first principles as follows. Derivation of the Differential Form Suppose first an amount of quantity q is contained in a region of volume V, bounded by a closed surface S, as described above. This is equal to the amount already in V, plus the generated amount s (total - not per unit time or volume): (t’ is just a dummy variable of the generation integral). The rate of change of q leaving the region is simply the time derivative:

where the minus sign has been inserted since the amount of q is decreasing in the region. (Partial derivatives are used since they enter the integrand, which is not only a function of time, but also space due to the density nature of cp - differentiation needs only to be with respect to t). The rate of change of q crossing the boundary and leaving the region is:

so equating these expressions:

Using the divergence theorem on the left-hand side:

This is only true if the integrands are equal, which directly leads to the differential continuity equation:

Either form may be useful and quoted, both can appear in hydrodynamics and electromagnetism, but for quantum mechanics and energy conservation, only the first may be used. Therefore the first is more general. Equivalence Between Differential and Integral Form Starting from the differential form which is for unit volume, multiplying throughout by the infinitesimal volume element dV and integrating over the region gives the total amounts quantities in the volume of the region (per unit time):

again using the fact that V is constant in shape for the calculation, so it is independent of time and the time derivatives can be freely moved out of that integral, ordinary derivatives replace partial derivatives since the integral becomes a function of time only (the integral is evaluated

over the region - so the spatial variables become removed from the final expression and t remains the only variable). Using the divergence theorem on the left side

which is the integral form. Equivalence Between Elementary and Integral Form Starting from the surfaces are equal (since there is only one closed surface), so S1 = S2 = S and we can write: The left hand side is the flow rate of quantity q occurring inside the closed surface S. This must be equal to

since some is produced by sources, hence the positive term Σ , but some is also leaking out by passing through the surface, implied by the negative term -dq/dt. Similarly the right hand side is the amount of flux passing through the surface and out of it, so Equating these:

which is the integral form again. Electromagnetism 3-currents: In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing (local) charge conservation, or can be derived as a consequence of two of Maxwell’s equations. It states that the divergence of the current density J (in amperes per square metre) is equal to the negative rate of change of the charge density ρ (in coulombs per cubic metre), Maxwell’s equations are a quick way to obtain the continuity of charge.

Consistency with Maxwell’s Equations 4-currents Conservation of a current (not necessarily an electromagnetic current) is expressed compactly as the Lorentz invariant divergence of a four-current:

where • c is the speed of light • ρ; the charge density • j the conventional 3-current density. • μ; labels the space-time dimension since then which implies that the current is conserved:

Interpretation Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge. Fluid Dynamics In fluid dynamics, the continuity equation states that, in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff s current law in electric circuits. The differential form of the continuity equation is: where • ρ is fluid density, • t is time, • u is the flow velocity vector field. If ρ is a constant, as in the case of incompressible flow, the mass continuity equation simplifies to a volume continuity equation: which means that the divergence of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero. Further, the Navier-Stokes equations form a vector continuity equation describing the conservation of linear momentum. Energy Conservation of energy (which, in non-relativistic situations, can only be transferred, and not created or destroyed) leads to a continuity equation, an alternative mathematical statement of energy conservation to the thermodynamic laws. Letting:

• u =local energy density (energy per unit volume), • q = energy flux (transfer of energy per unit cross-sectional area per unit time) as a vector, the continuity equation is:

Quantum Mechanics In quantum mechanics, the conservation of probability also yields a continuity equation. The terms in the equation require these definitions, and are slightly less obvious than the other forms of volume densities, currents, current densities etc., so they are outlined here: •

The wavefunction ffor a single particle in the position-time space (rather than momentum space) - i.e. functions of position r and time t, Ψ = Ψ(r, t) = Ψ(x, y, z, t). • The probability density function ρ = ρ(r, t) is: • The probability that a measurement of the particle’s position will yield a value within V at t, denoted by P = Pr = V(t), • The probability current (aka probability flux) j:

With these definitions the continuity equation reads:

Either form is usually quoted. Intuitively; the above quantities indicate this represents the flow of probability. The chance of finding the particle at some r t flows like a fluid, the particle itself does not flow deterministically in the same vector field.

Consistency with Schrödinger’s Equation The 3-d time dependent Schrödinger equation and its complex conjugate (i → –i) throughout are respectively:

where U is the potential function. The partial derivative of p with respect to t is:

Multiplying the Schrödinger equation by Ψ* then solving for Ψ*∂Ψ/∂t, and similarly multiplying the complex conjugated Schrödinger equation by Ψ then solving for Ψ∂Ψ* / ∂t ;

substituting into the time derivative of

The Laplacian operators (Δ2) in the above result suggest that the right hand side is the divergence of j, and the reversed order of terms imply this is the negative of j, altogether:

so the continuity equation is:

The integral form follows as for the general equation. Consistency with the Wavefunction Probability Distribution The time derivative of P is

where the last equality follows from the product rule and the fact that the shape of V is fixed for the calculation and therefore independent of time - i.e. the time derivative can be moved through the integral. To simplify this further consider again the time dependent Schrödinger equation and its complex conjugate, in terms of the time derivatives of fand ΨΨ* respectively:

Substituting into the preceding equation:

From the product rule for the divergence operator

Substituting:

On the right side, the argument of the divergence operator is j,

using the divergence theorem again gives the integral form:

To obtain the differential form:

The differential form follows from the fact that the preceding equation holds for all V, and as the integrand is a continuous function of space, it must vanish everywhere:

Inverse-square Laws Any inverse-square law can instead be written in a Gauss’ law-type form (with a differential and integral form, as described above). Two examples are Gauss’ law (in electrostatics), which follows from the inverse-square Coulomb’s law, and Gauss’ law for gravity, which follows from the inverse-square Newton’s law of universal gravitation. The derivation of the Gauss’ law-type equation from the inverse-square formulation (or viceversa) is exactly the same in both cases. History The theorem was first discovered by Lagrange in 1762, then later independently rediscovered by Gauss in 1813, by Green in 1825 and in 1831 by Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the divergence theorem are correctly called Ostrogradsky’s theorem, but also commonly Gauss’s theorem, or Green’s theorem. Examples To verify the planar variant of the divergence theorem for a region R, where F(x, y) = 2yi + 5xj, and R is the region bounded by the circle x2+y2 = 1. The boundary of R is the unit circle, C, that can be represented parametrically by: x= cos(s); y= sin(s) such that 0 ≤ s ≤ 2π where s units is the length arc from the point s = 0 to the point P on C. Then a vector equation of C is C(s) = cos(s)i + sin(s)j. At a point P on C: Therefore,

Because

and because

. Thus

Generalization to Tensor Fields Writing the theorem in index notation:

suggestively, replacing the vector field F with a rank-n tensor field T, this can be generalized to:

where on each side, tensor contraction occurs for at least one index. This form of the theorem is still in 3d, each index takes values 1, 2, and 3. It can be generalized further still to higher dimensions (for example to 4d spacetime in general relativity).

Advection In chemistry, engineering and earth sciences, advection is a transport mechanism of a substance or conserved property by a fluid due to the fluid’s bulk motion. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air. In general, any substance or conserved, extensive quantity can be advected by a fluid that can hold or contain the quantity or substance. In advection, a fluid transports some conserved quantity or material via bulk motion. The fluid’s motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Advection requires currents in the fluid, and so cannot happen in rigid solids. It does not include transport of substances by simple diffusion. Advection is sometimes confused with the more encompassing process of convection. In fact, convective transport is the sum of advective transport and diffusive transport. In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle. Distinction Between Advection and Convection The term advection is sometimes used as a synonym for convection. Technically, convection is the sum of transport by diffusion and advection. Advective transport describes

the movement of some quantity via the bulk flow of a fluid (as in a river or pipeline). Meteorology In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle. Other Quantities The advection equation also applies if the quantity being advected is represented by a probability density function at each point, although accounting for diffusion is more difficult. Mathematics of Advection The advection equation is the partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. It is derived using the scalar field’s conservation law, together with Gauss’s theorem, and taking the infinitesimal limit. One easily-visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a “pulse” via advection, as the water’s movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a diffusive manner, which is not advection. Note that as it moves downstream, the “pulse” of ink will also spread via diffusion. The sum of these processes is called convection. The Advection Equation In Cartesian coordinates the advection operator is

where u = (ux, uy, uz) is the velocity field, and Δ is the del operator (note that Cartesian coordinates are used here). The advection equation for a conserved quantity described by a scalar field Ψ is expressed mathematically by a continuity equation: where Δ is the divergence operator and again u is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies Δ·u=0 and u is said to be solenoidal. If this is so, the above equation reduces to In particular, if the flow is steady, then which shows that Ψ is constant along a streamline. If a vector quantity a (such as a magnetic field) is being advected by the solenoidal velocity field u, the advection equation above

becomes: Here, a is a vector field instead of the scalar field y}. Solving the Equation The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centres on discontinuous “shock” solutions (which are notoriously difficult for numerical schemes to handle). Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes where Ψ = Ψ(x, t) is the scalar field being ux is the x component of the vector u = (ux, 0,0). According to Zang, numerical simulation can be aided by considering the skew symmetric form for the advection operator. where and u is the same as above. Since skew symmetry implies only imaginary eigenvalues, this form reduces the “blow up” and “spectral blocking” often experienced in numerical solutions with sharp discontinuities. Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems.

This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges.

Beltrami Vector Field In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that F×(∇×F) = 0. If F is solenoidal - that is, if ∇.F = 0 such as for an incompressible fluid or a magnetic field, we may examine ∇×(∇×F) ≡ ∇2F-∇∇.F and use apply this identity twice to find that

∇2F = ∇×(λF) and if we further assume that λis a constant, we arrive at the simple form ∇2F = λ2F. Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions. The vector field

is a multiple of the standard contact structure -zi + j, and furnishes an example of a Beltrami vector field.

Bivector In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities.

Figure: Parallel plane segments with the same orientation and area corresponding to the same bivector a ≠ b.

Bivectors are generated by the exterior product on vectors – given two vectors a and b their exterior product a ≠ b is a bivector. But not all bivectors can be generated this way, and in higher dimensions a sum of exterior products is often needed. More precisely a bivector that requires only a single exterior product is simple; in two and three dimensions all bivectors are simple, but in higher dimensions this is not generally the case. The exterior product is antisymmetric, so b ≠ a is the negation of the bivector a ≠ b, producing a rotation with the opposite sense, and a ≠ a is the zero bivector. Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. Specifically for the bivector a ≠ b, its magnitude is the area of the parallelogram with edges a and b, its attitude that of any plane specified by a and b, and its orientation the sense of the rotation that would align a with b. It does not have a definite location or position.

History The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra, as the result of the exterior product. Around the same time in 1843 in Ireland William Rowan Hamilton discovered quaternions. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann’s algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector as it is known today was fully understood. Around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus which included separate cross product and dot products, derived from quaternion multiplication. The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, meant the insights of Hamilton and Clifford were overlooked for a long time, as much of 20th century mathematics and physics was formulated in vector terms. Gibbs instead described bivectors as vectors, and used “bivector” to describe an unrelated quantity, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra, a more restricted Clifford algebra over real or complex vector spaces with nondegenerate quadratic form. Its resurgence was led by David Hestenes who, along with others, discovered a range of new applications in physics for geometric algebra. Formal Definition For this article the bivector will be considered only in real geometric algebras. This in practice is not much of a restriction, as all useful applications are drawn from such algebras. Also unless otherwise stated, all examples have a Euclidean metric and so a quadratic form with signature 1. Geometric Algebra and the Geometric Product The bivector arises from the definition of the geometric product over a vector space. For vectors a, b and c, the geometric product on vectors is defined as follows: Associativity: (ab)c = a(bc) Left and right distributivity: a(b + c) = ab + ac (b + c)a = ba + ca Contraction: a2 = Q(a)=∞a|a|2 Where Q is the quadratic form, | a | is the magnitude of a and υa is the signature of the vector. For a space with Euclidean metric υa is 1 so can be omitted, and the contraction condition becomes: a2 = |a|2

The Interior Product From associativity a(ab) = a2b, a scalar times b. When b is not parallel to and hence not a scalar multiple of a, ab cannot be a scalar. But

is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is |a||b|cosθ, where θ is the angle between the vectors. It is therefore identical to the interior product between two vectors, and is written the same way,

It is symmetric, scalar valued, and can be used to determine the angle between two vectors: in particular if a and b are orthogonal the product is zero. The Exterior Product In the same way another quantity can be written down: This is called the exterior product, a ≠ b. It is antisymmetric in a and b, that is

By addition:

That is the geometric product is the sum of the symmetric interior product and antisymmetric exterior product. To calculate a ≠ b consider the sum Expanding using the geometric product and simplifying gives So using the Pythagorean trigonometric identity: With a negative square it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude |a||b|sinθ, where θ is the angle between the vectors, and so is zero for parallel vectors. To distinguish them from vectors, bivectors are written here with bold capitals, for example: A = a ∧ b = -b ∧a, although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra.

Properties The space ∧2ℝn: the algebra generated by the geometric product is the geometric algebra over the vector space. For a Euclidean vector space it is written Gn or C-£n(R), where n is the dimension of the vector space ℝn. C-£n is both a vector space and an algebra, generated by all the products between vectors in ℝn, so it contains all vectors and bivectors. More precisely as a vector space it contains the vectors and bivectors as subspaces. The space of all bivectors is written ∧2ℝn. Unlike ℝn it is not a Euclidean subspace; nor is it a subalgebra. The Even Subalgebra The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written C- £ + n . This algebra results from considering all products of scalars and bivectors generated by the geometric product. It has dimension 2n 1, and contains ∧2ℝn as a linear subspace with dimension 1D2 n(n - 1) (a triangular number). In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest. In two dimensions the even subalgebra is isomorphic to the complex numbers, l, while in three it is isomorphic to the quaternions, l More generally the even subalgebra can be used to generate rotations in any dimension, and can be generated by bivectors in the algebra. Magnitude As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectors a and b, is |a||b|sin θ, where θ is the angle between the vectors. It is written |B|, where B is the bivector. For general bivectors the magnitude can be calculated by taking the norm of the bivector considered as a vector in the space ∧a2ℝn. If the magnitude is zero then all the bivector’s components are zero, and the bivector is the zero bivector which as an element of the geometric algebra equals the scalar zero. Unit Bivectors A unit bivector is one with unit magnitude. It can be derived from any non-zero bivector by dividing the bivector by its magnitude, that is

Of particular interest are the unit bivectors formed from the products of the standard basis. If ei and ej . are distinct basis vectors then the product ei ≠ ej is a bivector. As the vectors are orthogonal this is just e.e., written e.., with unit magnitude as the vectors are unit vectors. The set of all such bivectors form a basis for ∧2ℝn. For instance in four dimensions the basis for ∧ℝ4 is (e1e2, e1e3, e1e4, e2e3, e2e4, e3e4) or (e12, e13, e14, e23, e24, e34). Simple Bivectors

The exterior product of two vectors is a bivector, but not all bivectors are exterior products of two vectors. For example in four dimensions the bivector cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by a oriented plane area. Product of two Bivectors The geometric product of two bivectors, A and B, is The quantity A . B is the scalar valued interior product, while A ≠ B is the grade 4 exterior product that arises in four or more dimensions. The quantity A × B is the bivector valued commutator product, given by

The space of bivectors ∧2ℝn are a Lie algebra over ℝ, with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra. Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to If the bivector is simple the last term is zero and the product is the scalar valued A · A, which can be used as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple. Two Dimensions When working with coordinates in geometric algebra it is usual to write the basis vectors as (e1, e2, ...), a convention that will be used here. A vector in real two dimensional space ℝ2 can be written a = a1e1 + a2e2, where a1 and a2 are real numbers, e1 and e2 are orthonormal basis vectors. The geometric product of two such vectors is

This can be split into the symmetric, scalar valued, interior product and an antisymmetric, bivector valued exterior product:

All bivectors in two dimensions are of this form, that is multiples of the bivector e1e2, written e12 to emphasise it is a bivector rather than a vector. The magnitude of e12 is 1, with e122 = -1, so it is called the unit bivector. The term unit bivector can be used in other dimensions

but it is only uniquely defined in two dimensions and all bivectors are multiples of e12. As the highest grade element of the algebra e12 is also the pseudoscalar which is given the symbol i. Complex Numbers With the properties of negative square and unit magnitude the unit bivector can be identified with the imaginary unit from complex numbers. The bivectors and scalars together form the even subalgebra of the geometric algebra, which is isomorphic to the complex numbers c . The even subalgebra has basis (1, e12), the whole algebra has basis (1, e1, e2, e12). The complex numbers are usually identified with the coordinate axes and two dimensional vectors, which would mean associating them with the vector elements of the geometric algebra. There is no contradiction in this, as to get from a general vector to a complex number an axis needs to be identified as the real axis, e1 say. This multiplies by all vectors to generate the elements of even subalgebra. All the properties of complex numbers can be derived from bivectors, but two are of particular interest. First as with complex numbers products of bivectors and so the even subalgebra are commutative. This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions. Second a general bivector can be written θe12 = iθ, where θ is a real number. Putting this into the Taylor series for the exponential map and using the property e122 = -1 results in a bivector version of Euler’s formula, which when multiplied by any vector rotates it through an angle θ about the origin: The product of a vector with a bivector in two dimensions is anticommutative, so the following products all generate the same rotation Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol R, so in two dimensions a rotor that rotates through angle θ can be written and the rotation it generates is Three Dimensions In three dimensions the geometric product of two vectors is

This can be split into the symmetric, scalar valued, interior product and the antisymmetric, bivector valued, exterior product:

In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors e23, e31 and e12 form a basis for the space of bivectors ∧2ℝ3, which itself a three dimensional linear space. So if a general bivector is: they can be added like vectors while when multiplied they produce the following which can be split into symmetric scalar and antisymmetric bivector parts as follows

The exterior product of two bivectors in three dimensions is zero. A bivector B can be written as the product of its magnitude and a unit bivector, so writing β for |B| and using the Taylor series for the exponential map it can be shown that

This is another version of Euler’s formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example in general eA + B ≠ eAeB in three (or more) dimensions. The full geometric algebra in three dimensions, C–3(ℝ), has basis (1, e1, e2, e3, e23, e31, e12, e123). The element e123 is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors to which they are related, as discussed below. Quaternions Bivectors are not closed under the geometric product, but the even subalgebra is. In three dimensions it consists of all scalar and bivector elements of the geometric algebra, so a general element can be written for example a + A, where a is the scalar part and A is the bivector part. It is written C-l + 3 and has basis (1, e23, e31, e12). The product of two general elements of the even subalgebra is (a + A)(b + B) = ab + aB + bA + A-B + A×B. The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, H. This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates

to the negative products in the bivector interior product A · B. Other quaternion properties can be similarly related to or derived from geometric algebra. This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done there is no special quaternion product, there is just the normal geometric product on the elements. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions. Rotation Vector The rotation vector, from the axis angle representation of rotations, is a compact way of representing rotations in three dimensions. In its most compact form it consists of a vector, the product of the a unit vector that is the axis of rotation and the angle of rotation, so the magnitude of the vector is the rotation angle. In geometric algebra this vector is classified as a bivector. This can be seen in its relation to quaternions. If the axis is ω and the angle of rotation is θ then the rotation vector is ωθ quaternion associated with the rotation is

but this is just the exponent of half of the bivector Φθ, that is

So rotation vectors are bivectors, just as quaternions are elements of the geometric algebra, and they are related by the exponential map in that algebra. Rotors The bivector Φθ generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions: As to two dimensions the quantity eΦθ is called a rotor and written R. The quantity e-Φθ is then R-1, and they generate rotations as follows This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors R and -R represent the same rotation. Matrices Bivectors are isomorphic to skew-symmetric matrices; the general bivector B23e23 + B31e31 + B12e12 maps to the matrix

This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector; an example is the angular velocity tensor. Skew symmetric matrices generate orthogonal matrices with determinant 1 through the exponential map. In particular the exponent of a bivector associated with a rotation is a rotation matrix, that is the rotation matrix MR given by the above skew-symmetric matrix is MR =eMB . The rotation described by MR is the same as that described by the rotor R given by and the matrix MR can be also calculated directly from rotor R:

Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix M the eigenvalues can calculated by solving the characteristic equation for that matrix 0 = det (M - λI). By the fundamental theorem of algebra this has three roots, but only one real root as there is only one eigenvector, the axis of rotation. The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors result in the bivector, or at least a multiple of it. Axial Vectors The rotation vector is an example of an axial vector. Axial vectors or pseudovectors are vectors that undergo a sign change compared to normal or polar vectors under inversion, that is when reflected or otherwise inverted. Examples include quantities like torque, angular momentum and vector magnetic fields. Such quantities can be described as bivectors in geometric algebra; that is quantities that might use axial vectors in vector algebra are better represented by bivectors in geometric algebra. More precisely, the Hodge dual gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice-versa; that is A = *a; a = *A where * indicates the Hodge dual. Alternately, using the unit pseudoscalar in C-l3(ℝ), i = e1e2e3 gives A = ai; a = -Ai. This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar i squares to -1, so a negative is needed in one of the products. This relationship extends to operations like the vector valued cross product and bivector valued exterior product, as when written as determinants they are calculated in the same way:

so are related by the Hodge dual: *(a∧b) = a×b ; *(a×b) = a∧b. Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalises to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways. Geometric Interpretation As suggested by their name and that of the algebra, one of the attractions of bivectors is that they have a natural geometric interpretation. This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions. In two dimensions the geometric interpretation is trivial, as the space is two dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor.

Figure: Parallel plane segments with the same orientation and area corresponding to the same bivector a ≠ b.

All bivectors can be interpreted as planes, or more precisely as directed plane segments. In three dimensions there are three properties of a bivector that can be interpreted geometrically: •

The arrangement of the plane in space, precisely the attitude of the plane (or alternately the rotation, geometric orientation or gradient of the plane), is associated with the ratio of the bivector components. In particular the three basis bivectors, e23, e31 and e12, or scalar multiples of them, are associated with the yz-plane, xz-plane and xy-plane respectively. • The magnitude of the bivector is associated with the area of the plane segment. The area does not have a particular shape so any shape can be used. It can even be

represented in other ways, such as by an angular measure. But if the vectors are interpreted as lengths the bivector is usually interpreted as an area with the same units, as follows. • Like the direction of a vector a plane associated with a bivector has a direction, a circulation or a sense of rotation in the plane, which takes two values seen as clockwise and counterclockwise when viewed from viewpoint not in the plane. This is associated with a change of sign in the bivector, that is if the direction is reversed the bivector is negated. Alternately if two bivectors have the same attitude and magnitude but opposite directions then one is the negative of the other.

Figure: The cross product a × b is orthogonal to the bivector a ≠ b.

In three dimensions all bivectors can be generated by the exterior product of two vectors. If the bivector B = a ≠ b then the magnitude of B is where θ is the angle between the vectors. This is the area of the parallelogram with edges a and b. One interpretation is that the area is swept out by b as it moves along a. The exterior product is antisymmetric, so reversing the order of a and b to make a move along b results in a bivector with the opposite direction that is the negative of the first. The plane of bivector a ≠ b contains both a and b so they are both parallel to the plane. Bivectors and axial vectors are related by Hodge dual. In a real vector space the Hodge dual relates a subspace to its orthogonal complement, so if a bivector is represented by a plane then the axial vector associated with it is simply the plane’s surface normal. The plane has two normals, one on each side, giving the two possible orientations for the plane and bivector. This relates the cross product to the exterior product. It can also be used to represent physical quantities, like torque and angular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of the force, linear momentum or displacement that they are calculated from. But if a bivector is used instead the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way they act. It also unlike the vector representation generalises into other dimensions.

Figure: Relationship between force F, torque τ, linear momentum p, and angular momentum L.

The product of two bivectors has a geometric interpretation. For non-zero bivectors A and B the product can be split into symmetric and antisymmetric parts as follows: AB = A·B + A × B. Like vectors these have magnitudes |A · B| = |A||B| cos θ and |A × B| = |A||B| sin θ, where θ is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions.

Figure: Two bivectors, two of the non-parallel sides of a prism, being added to give a third bivector.

Bivectors can be added together as areas. Given two nonzero bivectors B and C in three dimensions it is always possible to find a vector that is contained in both, a say, so the bivectors can be written as exterior products involving a: B = a∧b C = a ∧c This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of a prism with a, b, c and b + c as edges. This corresponds to the two ways of calculating the area using the distributivity of the exterior product:

This only works in three dimensions as it is the only dimension where a vector parallel to both bivectors must exist. In higher dimensions bivectors generally are not associated with a single plane, or if they are (simple bivectors) two bivectors may have no vector in common, and so sum to a non-simple bivector. Four Dimensions In four dimensions the basis elements for the space ∧2 r 4 of bivectors are (e12, e13, e14, e23, e24, e34), so a general bivector is of the form

Orthogonality In four dimensions bivectors are orthogonal to bivectors. That is the dual of a bivector is a bivector, and the space ∧2ℝ4 is dual to itself in C-l4(ℝ). Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its dual space. This can be used to partition the bivectors into two ‘halves’, for example into two sets of three unit bivectors each. There are only four distinct ways to do this, and whenever it’s done one vector is in only one of the two halves, for example (e12, e13, e14) and (e23, e24, e34). Simple Bivectors in 4D In four dimensions bivectors are generated by the exterior product of vectors in ℝ4, but with one important difference from ℝ3 and R2. In four dimensions not all bivectors are simple. There are bivectors such as e12 + e34 that cannot be generated by the external product of two vectors. This also means they do not have a real, that is scalar, square. In this case The element e1234 is the pseudoscalar in C£ 4, distinct from the scalar, so the square is non-scalar. All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as Alternately every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover for a general bivector the choice of simple bivectors is unique, that is there is only one way to decompose into orthogonal bivectors. This is true also for simple bivectors, except one of the orthogonal parts is zero. The exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique. Rotations in ℝ4 As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions if B is a bivector then the rotor R is eB/2 and rotations are generated in the same way:

Figure: A 3D projection of an tesseract performing an isoclinic rotation.

The rotations generated are more complex though. They can be categorised as follows: • simple rotations are those that fix a plane in 4D, and rotate by an angle “about” this plane. • double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified • isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique. These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique. Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector B = B1 + B2, where B1 and B2 are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows: It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors. Spacetime Rotations Spacetime is a mathematical model for our universe used in special relativity. It consists of three space dimensions and one time dimension combined into a single four dimensional space. It is naturally described if using geometric algebra and bivectors, with the Euclidean metric replaced by a Minkowski metric. That is the algebra is identical to that of Euclidean space, except the signature is changed, so

(Note the order and indices above are not universal – here e4 is the time-like dimension). The geometric algebra is C– £31(ℝ), and the subspace of bivectors is ∧2ℝ31. The bivectors are of two types. The bivectors e23, e31 and e12 have negative squares and correspond to the bivectors of the three dimensional subspace corresponding to Euclidean space, ℝ3. These bivectors generate normal rotations in ℝ3. The bivectors e14, e24 and e34 have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions hyperbolic functions are needed, which generates a rotor as follows:

These are Lorentz transformations, expressed in a particularly compact way, using the same algebra as in ℝ3 and ℝ4. In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector A is of the form The set of all rotations in spacetime form the Lorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same algebra. Maxwell’s Equations (Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors J and A exceptionally in uppercase) Maxwell’s equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from ∧2ℝ31. If the electric and magnetic fields in ℝ3 are E and B then the electromagnetic bivector is

where e4 is again the basis vector for the time-like dimension and c is the speed of light. The quantity Be123 is the bivector dual to B in three dimensions, as discussed above, while Ee4 as a product of orthogonal vectors is also bivector valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current J, a vector quantity given by where j is current density and ρ is charge density. They are related by a differential operator ∂, which is

The operator Δ is a differential operator in geometric algebra, acting on the space dimensions and given by Δ M = Δ M + Δ ≤ M. When applied to vectors Δ M is the divergence and Δ · M is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity M they act as grade lowering and raising differential operators. In particular if M is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator. Together these can be used to give a particularly compact form for Maxwell’s equations in a vacuum: ∂F = J. This when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell’s four

equations. This is the form in a vacuum, but the general form is only a little more complex. It is also related to the electromagnetic four-potential, a vector A given by

where A is the vector magnetic potential and V is the electric potential. It is related to the electromagnetic bivector as follows ∂A = -F, using the same differential operator. Higher Dimensions As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real space ℝn is C^n(ℝ), and the subspace of bivectors is ∧2 ℝ n. The number of simple bivectors needed to form a general bivector rises with the dimension, so for n odd it is (n - 1) / 2, for n even it is n / 2. So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions. For example in six dimensions with standard basis (e1, e2, e3, e4, e5, e6) the bivector e12 + e34 + e56 is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum. Rotations in Higher Dimensions As in three and four dimensions rotors are generated by the exponential map, so is the rotor generated by bivector B. Simple rotations, that take place in a plane of rotation around a fixed blade of dimension (n - 2) are generated by with simple bivectors, while other bivectors generate more complex rotations which can be described in terms of the simple bivectors they are sums of, each related to a plane of rotation. All bivectors can be expressed as the sum of orthogonal and commutative simple bivectors, so rotations can always be decomposed into a set of commutative rotations about the planes associated with these bivectors. The group of the rotors in n dimensions is the spin group, Spin(n). One notable feature, related to the number of simple bivectors and so rotation planes, is that in odd dimensions every rotation has a fixed axis - it is misleading to call it an axis of rotation as in higher dimensions rotations are taking place in multiple planes orthogonal to it. This is related to bivectors, as bivectors in odd dimensions decompose into the same number of bivectors as the even dimension below, so have the same number of planes, but one extra dimension. As each plane generates rotations in two dimensions in odd dimensions there must be one dimension, that is an axis, that is not being rotated. Bivectors are also related to the rotation matrix in n dimensions. As in three dimensions the characteristic equation of the matrix can be solved to find the eigenvalues. In odd dimensions this has one real root, with eigenvector the fixed axis, and in even dimensions it has no real roots, so either all or all but one of the roots are complex conjugate pairs. Each

pair is associated with a simple component of the bivector associated with the rotation. In particular the log of each pair is ± the magnitude, while eigenvectors generated from the roots are parallel to and so can be used to generate the bivector. In general the eigenvalues and bivectors are unique, and the set of eigenvalues gives the full decomposition into simple bivectors; if roots are repeated then the decomposition of the bivector into simple bivectors is not unique. Projective Geometry Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is Ci n(ℝ), n > 3, the algebra of the real vector space ℝn. This is used to describe objects in the real projective space ℝ → n-1. The nonzero vectors in Cln( ℝ) or ℝn are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in ∧2ℝn represent lines in ℝ→n-1, with bivectors differing only by a (positive or negative) scale factor representing the same line. A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example given two distinct points in ℝ→n-1represented by vectors a and b the line between them is given by a*b (or b*a). Two lines intersect in a point if A ≠ B = 0 for their bivectors A and B. This point is given by the vector The operation “A” is the meet, which can be defined as above in terms of the join, J = A ≠ B for non-zero A ≠ B. Using these operations projective geometry can be formulated in terms of geometric algebra. For example given a third (non-zero) bivector C the point p lies on the line given by C if and only if So the condition for the lines given by A, B and C to be collinear is which in Cl3(ℝ) and ℝ→2 simplifies to 〈ABC〉 = 0, where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.

Comparison of Vector Algebra and Geometric Algebra Vector algebra and geometric algebra are alternative approaches to providing additional algebraic structures on vector spaces, with geometric interpretations, particularly vector fields in multivariable calculus and applications in mathematical physics. Vector algebra is simpler, but specific to Euclidean 3-space, while geometric algebra uses multilinear algebra, but works in all dimensions and signatures, notably 3+1 spacetime, as well as 2 dimensions.

They are mathematically equivalent in 3 dimensions, though the approaches differ. Vector algebra is more widely used in elementary multivariable calculus, while geometric algebra is used in some more advanced treatments, and is proposed for elementary use as well. In advanced mathematics, particularly differential geometry, neither is widely used, with differential forms being far more widely used.

Basic Concepts and Operations In vector algebra the basic objects are scalars and vectors, and the operations (beyond the vector space operations of scalar multiplication and vector addition) are the dot (or scalar) product and the cross product ×. In geometric algebra the basic objects are multivectors (scalars are 0-vectors, vectors are 1-vectors, etc.), and the operations include the Clifford product (here called “geometric product”) and the exterior product. The dot product/inner product/scalar product is defined on 1-vectors, and allows the geometric product to be expressed as the sum of the inner product and the exterior product when multiplying 1-vectors. A distinguishing feature is that vector algebra uses the cross product, while geometric algebra uses the exterior product (and the geometric product). More subtly, geometric algebra in Euclidean 3-space distinguishes 0-vectors, 1-vectors, 2-vectors, and 3-vectors, while elementary vector algebra identifies 1-vectors and 2-vectors (as vectors) and 0-vectors and 3-vectors (as scalars), though more advanced vector algebra distinguishes these as scalars, vectors, pseudovectors, and pseudoscalars. Unlike vector algebra, geometric algebra includes sums of k-vectors of differing k. The cross product does not generalize to dimensions other than 3 (as a product of two vectors, yielding a third vector), and in higher dimensions not all k-vectors can be identified with vectors or scalars. By contrast, the exterior product (and geometric product) is defined uniformly for all dimensions and signatures, and multivectors are closed under these operations. Embellishments, ad Hoc Techniques, and Tricks More advanced treatments of vector algebra add embellishments to the initial picture – pseudovectors and pseudoscalars (in geometric algebra terms, 2-vectors and 3-vectors), while applications to other dimensions use ad hoc techniques and “tricks” rather than a general mathematical approach. By contrast, geometric algebra begins with a complete picture, and applies uniformly in all dimensions. For example, applying vector calculus in 2 dimensions, such as to compute torque or curl, requires adding an artificial 3rd dimension and extending the vector field to be constant in that dimension. The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines torque and curl as pseudoscalar fields (2vector fields), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product. List of Analogous Formulas Here are some comparisons between standard ℝ3 vector relations and their

corresponding wedge product and geometric product equivalents. All the wedge and geometric product equivalents here are good for more than three dimensions, and some also for two. In two dimensions the cross product is undefined even if what it describes (like torque) is perfectly well defined in a plane without introducing an arbitrary normal vector outside of the space. Many of these relationships only require the introduction of the wedge product to generalize, but since that may not be familiar to somebody with only a traditional background in vector algebra and calculus, some examples are given. Algebraic and Geometric Properties of Cross and Wedge Products Cross and wedge products are both antisymmetric:

They are both linear in the first operand

and in the second operand

In general, the cross product is not associative, while the wedge product is

Both the cross and wedge products of two identical vectors are zero: u×u = 0 u∧u = 0 u×v is perpendicular to the plane containing u and v. u∧ v is an oriented representation of the same plane. The cross product of traditional vector algebra (on ℝ3) find its place in geometric algebra G3 as a scaled exterior product a × b = -i(a∧b) (this is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the distinction between vectors and bivectors (elements of grade two). The here is a unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property The equivalence of the ℝ3 cross product and the wedge product expression above can be confirmed by direct multiplication of-i = -e1e2e3 with a determinant expansion of the wedge product

Essentially, the geometric product of a bivector and the pseudoscalar of Euclidean 3space provides a method of calculation of the Hodge dual. Norm of a Vector The norm (length) of a vector is defined in terms of the dot product ||u||2= u·u Using the geometric product this is also true, but this can be also be expressed more compactly as ||u||2 = u2 This follows from the definition of the geometric product and the fact that a vector wedge product with itself is zero uu = u·u + u∧u = u-u Lagrange Identity In three dimensions the product of two vector lengths can be expressed in terms of the dot and cross products The corresponding generalization expressed using the geometric product is This follows from expanding the geometric product of a pair of vectors with its reverse Determinant Expansion of Cross and Wedge Products

Without justification or historical context, traditional linear algebra texts will often define the determinant as the first step of an elaborate sequence of definitions and theorems leading up to the solution of linear systems, Cramer’s rule and matrix inversion. An alternative treatment is to axiomatically introduce the wedge product, and then demonstrate that this can be used directly to solve linear systems. This is shown below, and does not require sophisticated math skills to understand. It is then possible to define determinants as nothing more than the coefficients of the wedge product in terms of “unit k-vectors” (ei∧ej terms) expansions as above. • A one by one determinant is the coefficient of e1 for an jj1 1-vector. • A two-by-two determinant is the coefficient of e1 ∧ e2 for an jj2 bivector • A three-by-three determinant is the coefficient of e1 ∧ e2 ∧ e3 for an jj3 trivector... When linear system solution is introduced via the wedge product, Cramer’s rule follows

as a side effect, and there is no need to lead up to the end results with definitions of minors, matrices, matrix invertibility, adjoints, cofactors, Laplace expansions, theorems on determinant multiplication and row column exchanges, and so forth. Matrix Related Matrix inversion (Cramer’s rule) and determinants can be naturally expressed in terms of the wedge product. The use of the wedge product in the solution of linear equations can be quite useful for various geometric product calculations. Traditionally, instead of using the wedge product, Cramer’s rule is usually presented as a generic algorithm that can be used to solve linear equations of the form Ax = b (or equivalently to invert a matrix). Namely

This is a useful theoretic result. For numerical problems row reduction with pivots and other methods are more stable and efficient. When the wedge product is coupled with the Clifford product and put into a natural geometric context, the fact that the determinants are used in the expression of ℝN parallelogram area and parallelepiped volumes (and higher dimensional generalizations of these) also comes as a nice side effect. As is also shown below, results such as Cramer’s rule also follow directly from the property of the wedge product that it selects non identical elements. The end result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule. Two variables example

Pre and post multiplying by a and b

Provided aA ∧ b ≠ 0the solution is

For a, b ∈ ℝ2, this is Cramer’s rule since the e1 ∧ e2 factors of the wedge products

divide out. Similarly, for three, or N variables, the same ideas hold

Again, for the three variable three equation case this is Cramer’s rule since the e1 ∧ e2 ∧ e3 factors of all the wedge products divide out, leaving the familiar determinants. A numeric example with three equations and two unknowns When there are more equations than variables case, if the equations have a solution, each of the k-vector quotients will be scalars To illustrate here is the solution of a simple example with three equations and two unknowns.

The right wedge product with (1,1,1) solves for x

and a left wedge product with (1,1,0) solves for y

Observe that both of these equations have the same factor, so one can compute this only once (if this was zero it would indicate the system of equations has no solution). Collection of results for and yields a Cramers rule like form:

Writing ei ∧ ei = eij , we have the end result:

Equation of a Plane For the plane of all points r through the plane passing through three independent points r1, r0, and r2, the normal form of the equation is The equivalent wedge product equation is Projection and Rejection Using the Gram-Schmidt process a single vector can be decomposed into two components with respect to a reference vector, namely the projection onto a unit vector in a reference direction, and the difference between the vector and that projection. With, û = u/| |u| |, the projection of onto is Orthogonal to that vector is the difference, designated the rejection,

The rejection can be expressed as a single geometric algebraic product in a few different ways

The similarity in form between the projection and the rejection is notable. The sum of these recovers the original vector Here the projection is in its customary vector form. An alternate formulation is possible that puts the projection in a form that differs from the usual vector formulation

Working backwards from the end result, it can be observed that this orthogonal decomposition result can in fact follow more directly from the definition of the geometric product itself. With this approach, the original geometrical consideration is not necessarily obvious, but it is a much quicker way to get at the same algebraic result. However, the hint that one can work backwards, coupled with the knowledge that the wedge product can be used to solve sets of linear equations, the problem of orthogonal decomposition can be posed directly, Let v = au + x , where u-x = 0 . To discard the portions of that are collinear with , take the wedge product u∧v = u∧ (au + x) = u∧x Here the geometric product can be employed u∧v = u∧x = ux-u-x = ux Because the geometric product is invertible, this can be solved for x

The same techniques can be applied to similar problems, such as calculation of the component of a vector in a plane and perpendicular to the plane. For three dimensions the projective and rejective components of a vector with respect to an arbitrary non-zero unit vector, can be expressed in terms of the dot and cross product For the general case the same result can be written in terms of the dot and wedge product and the geometric product of that and the unit vector It’s also worthwhile to point out that this result can also be expressed using right or left vector division as defined by the geometric product

Like vector projection and rejection, higher dimensional analogs of that calculation are also possible using the geometric product. As an example, one can calculate the component of a vector perpendicular to a plane and the projection of that vector onto the plane. Let w = au + bv + x , where u-x = vx =0 . As above, to discard the portions of that are colinear with v or v, take the wedge product Having done this calculation with a vector projection, one can guess that this quantity equals x(u∧v). One can also guess there is a vector and bivector dot product like quantity such that the allows the calculation of the component of a vector that is in the “direction of a plane”. Both of these guesses are correct, and validating these facts is worthwhile. However, skipping ahead slightly, this to-be-proved fact allows for a nice closed form solution of the vector component outside of the plane:

Notice the similarities between this planar rejection result a the vector rejection result. To calculation the component of a vector outside of a plane we take the volume spanned by three vectors (trivector) and “divide out” the plane. Independent of any use of the geometric product it can be shown that this rejection in terms of the standard basis is

where

is the squared area of the parallelogram formed by , and . The (squared) magnitude of x is

Thus, the (squared) volume of the parallelopiped (base area times perpendicular height) is

Note the similarity in form to the w, u, v trivector itself

which, if you take the set of ei ∧ ej ∧ ek as a basis for the trivector space, suggests this is the natural way to define the length of a trivector. Loosely speaking the length of a vector is a length, length of a bivector is area, and the length of a trivector is volume. If a vector is factored directly into projective and rejective terms using the geometric then it is not necessarily obvious that the rejection term, a product of product vector and bivector is even a vector. Expansion of the vector bivector product in terms of the standard basis vectors has the following form

It can be shown that

(a result that can be shown more easily straight from r = v-û(û-v)). The rejective term is implies r · u =0. perpendicular to u, since The magnitude of r, is

So, the quantity

is the squared area of the parallelogram formed. It is also noteworthy that the bivector can be expressed as

Thus is it natural, if one considers each term ei ∧ ej as a basis vector of the bivector space, to define the (squared) “length” of that bivector as the (squared) area. we Going back to the geometric product expression for the length of the rejection see that the length of the quotient, a vector, is in this case is the “length” of the bivector divided by the length of the divisor. This may not be a general result for the length of the product of two ^-vectors, however it is a result that may help build some intuition about the significance of the algebraic operations. Namely, When a vector is divided out of the plane (parallelogram span) formed from it and another vector, what remains is the perpendicular component of the remaining vector, and its length is the planar area divided by the length of the vector that was divided out. Area of the Parallelogram Defined by u and V If A is the area of the parallelogram defined by u and v, then

and

Note that this squared bivector is a geometric multiplication; this computation can alternatively be stated as the Gram determinant of the two vectors. Angle Between two Vectors

Volume of the Parallelopiped Formed by Three Vectors In vector algebra, the volume of a parallelopiped is given by the square root of the squared norm of the scalar triple product:

Product of a Vector and a Bivector In order to justify the normal to a plane result above, a general examination of the product of a vector and bivector is required. Namely,

This has two parts, the vector part where i = j or i = k, and the trivector parts where no indexes equal. After some index summation trickery, and grouping terms and so forth, this is

The trivector term is w ∧u∧v. Expansion of (u ∧ v)w yields the same trivector term (it is the completely symmetric part), and the vector term is negated. Like the geometric product of two vectors, this geometric product can be grouped into symmetric and antisymmetric parts, one of which is a pure k-vector. In analogy the antisymmetric part of this product can be called a generalized dot product, and is roughly speaking the dot product of a “plane” (bivector), and a vector. The properties of this generalized dot product remain to be explored, but first here is a summary of the notation

Let w = x + y , where x = au + bv, and y-u = y-v = 0. Expressing w and the u∧v , products in terms of these components is w(u ∧v) = x(u ∧v) + y(u ∧ v) = x-(u ∧ v) + y-(u ∧ v) + y ∧ u ∧ v With the conditions and definitions above, and some manipulation, it can be shown that the term y-(u ∧ v) = 0 , which then justifies the previous solution of the normal to a plane problem. Since the vector term of the vector bivector product the name dot product is zero when the vector is perpendicular to the plane (bivector), and this vector, bivector “dot product” selects only the components that are in the plane, so in analogy to the vector-vector dot product this name itself is justified by more than the fact this is the non-wedge product term of the geometric vector-bivector product. Derivative of a Unit Vector It can be shown that a unit vector derivative can be expressed using the cross product

The equivalent geometric product generalization is

Thus this derivative is the component of in the direction perpendicular to r. In minus the projection of that vector onto r̂. other words this is This intuitively makes sense (but a picture would help) since a unit vector is constrained to circular motion, and any change to a unit vector due to a change in its generating vector has to be in the direction of the rejection of r̂ from . That rejection has to be scaled by 1/|r| to get the final result. When the objective isn’t comparing to the cross product, it’s also notable that this unit vector derivative can be written

Complex Lamellar Vector Field In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl. That is, F-(Δ × F) = 0. Complex lamellar vector fields are precisely those that are normal to a family of surfaces. A special case are irrotational vector fields, satisfying Δ × F = 0. An irrotational vector field is locally the gradient of a function, and is therefore

orthogonal to the family of level surfaces (the equipotential surfaces). Accordingly, the term lamellar vector field is sometimes used as a synonym for an irrotational vector field. The adjective “lamellar” derives from the noun “lamella”, which means a thin layer. The lamellae to which “lamellar flow” refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field.

Concatenation (Mathematics) In mathematics, concatenation is the joining of two numbers by their numerals. That is, the concatenation of 123 and 456 is 123456. Concatenation of numbers a and b is denoted a||b. Relevant subjects in recreational mathematics include Smarandache-Wellin numbers, home primes, and Champernowne’s constant. The convention for sequences at places such as the Online Encyclopedia of Integer Sequences is to have sequences of concatenations include as the first term a number prior to the actual act of concatenation. Therefore, care must be taken to ensure that parties discussing a topic agree either with this convention or with plain language. For example, the first term in the sequence of concatenations of increasing even numbers may be taken to be either 24, as would seem obviously correct, or simply 2, according to convention. Calculation The concatenation of numbers depends on the numeric base, which is often understood from context. Given the numbers p and q in base b, the concatenation p||q is given by

is the number of digits of q in base b, and [x] is the floor function. Vector Extension The concatenation of vectors can be understood in two distinct ways; either as a generalization of the above operation for numbers or as a concatenation of lists. Given two vectors in ℝn , concatenation can be defined as

In the case of vectors in jj1 , this is equivalent to the above definition for numbers. The further extension to matrices is trivial. Since vectors can be viewed in a certain way as lists, concatenation may take on another meaning. In this case the concatenation of two lists (a1, a2, ..., a n ) and (b1, b2, ..., bn ) is the list (a1, a2, ... a1n, b1, b2, ... bn ). Only the exact context will reveal which meaning is intended.

d’Alembert-Euler Condition In mathematics and physics, especially the study of mechanics and fluid dynamics, the d’Alembert-Euler condition is a requirement that the streaklines of a flow are irrotational. Let × = x(X,t) be the coordinates of the point × into which × is carried at time t by a (fluid) flow. be the second material derivative of x. Then the d’Alembert-Euler condition is: Let curl x = 0. The d’Alembert-Euler condition is named for Jean le Rond d’Alembert and Leonhard Euler who independently first described its use in the mid- 18th century. It is not to be confused with the Cauchy-Riemann conditions. Del In vector calculus, del is a vector differential operator, usually represented by the nabla symbol “. When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus. When applied to a field (a function defined on a multidimensional domain), del may denote the gradient (locally steepest slope) of a scalar field, the divergence of a vector field, or the curl (rotation) of a vector field, depending on the way it is applied. Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators, that makes many equations easier to write and remember. The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del “operator” with the field. These formal products do not necessarily commute with other operators or products. Definition In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del is defined in terms of partial derivative operators as

where {x̂ ŷ, ẑ} are the unit vectors in their respective directions. Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space Rn. In the Cartesian coordinate system with coordinates (x1, x2, ..., xn), del is:

where {êi : 1 ≤ i ≤ n} is the standard basis in this space. More compactly, using the Einstein summation notation, del is written as Notational Uses

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian. Gradient The vector derivative of a scalar field f is called the gradient, and it can be represented as:

It always points in the direction of greatest increase of f, and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane h(x,y), the 2d projection of the gradient at a given location will be a vector in the xy-plane (sort of like an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope. In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case: ∇(fg) = f ∇g + g∇ f However, the rules for dot products do not turn out to be simple, as illustrated by: Divergence The divergence of a vector field represented as:

a scalar function that can be

The divergence is roughly a measure of a vector field’s increase in the direction it points; but more accurately, it is a measure of that field’s tendency to converge toward or repel from a point. The power of the del notation is shown by the following product rule: The formula for the vector product is slightly less intuitive, because this product is not commutative: Curl The curl of a vector field

is a vector function that can be represented

as:

The curl at a point is proportional to the on-axis torque to which a tiny pinwheel would be subjected if it were centred at that point. The vector product operation can be visualized as a pseudo-determinant:

Again the power of the notation is shown by the product rule: Unfortunately the rule for the vector product does not turn out to be simple: Directional Derivative The directional derivative of a scalar field f(x,y,z) in the direction defined as:

This gives the change of a field f in the direction of a. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the “moving” derivative of the fluid. Laplacian The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; it is defined as:

The Laplacian is ubiquitous throughout modern mathematical physics, appearing in Laplace’s equation, Poisson’s equation, the heat equation, the wave equation, and the Schrödinger equation—to name a few. Tensor Derivative Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field vis a 9-term second-rank tensor, but can be denoted simply as Δ X v , where represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. For a small displacement §r, the change in the vector field is given by: Product Rules

Second Derivatives When del operates on a scalar or vector, generally a scalar or vector is returned.

Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:

Figure: DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles(dashed) mean that DD and GG do not exist.

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved, two of them are always zero:

Two of them are always equal: The 3 remaining vector derivatives are related by the equation: And one of them can even be expressed with the tensor product, if the functions are wellbehaved: Precautions Most of the above vector properties (except for those that rely explicitly on dels differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if del is replaced by any other vector. This is part of the tremendous value gained in representing this operator as a vector in its own right. Though you can often replace del with a vector and obtain a vector identity, making those identities intuitive, the reverse is not necessarily reliable, because del does not often commute. A counterexample that relies on del’s failure to commute:

A counter example that relies on del’s differential properties:

Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a precise numerical magnitude and direction, del does not have a precise value for either until it is allowed to operate on something. For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.

Field Line A field line is a locus that is defined by a vector field and a starting location within the field. Field lines are useful for visualizing vector fields, which are otherwise hard to depict. Note that, like longitude and latitude lines on a globe, or topographic lines on a topographic map, these lines are not physical lines that are actually present at certain locations; they are merely visualization tools.

Figure: Field lines depicting the electric field created by a positive charge (left), negative charge (centre), and uncharged object (right).

Figure: The figure at left shows the electric field lines of two equal positive charges. The figure at right shows the electric field lines of a dipole.

Figure: Iron filings arrange themselves so as to approximately depict some magnetic field lines. The magnetic field is created by a permanent magnet.

Precise Definition A vector field defines a direction at all points in space; a field line for that vector field may be constructed by tracing a topographic path in the direction of the vector field. More precisely, the tangent line to the path at each point is required to be parallel to the vector field at that point. A complete description of the geometry of all the field lines of a vector field is sufficient to completely specify the direction of the vector field everywhere. In order to also depict the magnitude, a selection of field lines is drawn such that the density of field lines (number of field lines per unit perpendicular area) at any location is proportional to the magnitude of the vector field at that point. As a result of the divergence theorem, field lines start at sources and end at sinks of the vector field. (A “source” is wherever the divergence of the vector field is positive, a “sink” is wherever it is negative.) In physics, drawings of field lines are mainly useful in cases where the sources and sinks, if any, have a physical meaning, as opposed to e.g. the case of a force field of a radial harmonic. For example, Gauss’s law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. (They can also potentially form closed loops, or extend to or from infinity). A gravitational field has no sources, it has sinks at masses, and it has neither elsewhere, gravitational field lines come from infinity and end at masses. A magnetic field has no sources or sinks (Gauss’s law for magnetism), so its field lines have no start or end: they can only form closed loops, or extend to infinity in both directions. Note that for these kinds of drawings, where the field-line density is intended to be proportional to the field magnitude, it is important to represent all three dimensions. For example, consider the electric field arising from a single, isolated point charge. The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. This means that their density is proportional to 1/ r2 , the correct result consistent with Coulomb’s law for this case. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to 1 / r , an incorrect result for this situation.

Chapter 4 Multipole Expansion A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be complex in general. Multipole expansions are very frequently used in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole, moment — is a constant, independent of angle. The following term — the first, or dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the quadrupole and octupole) vary more quickly with angles.

Expansion in Spherical Harmonics Most commonly, the series is written as a sum of spherical harmonics. Thus, we might write a function f(θ,ϕ) as the sum

Here, Ylm(θ,ϕ) are the standard spherical harmonics, and Clm are constant coefficients which depend on the function. The term C00 represents the monopole; C-1 , C10, C11 represent the dipole; and so on. Equivalently, the series is also frequently written as Here, each ni represents a unit vector in the direction given by the angles θ and ϕ, and indices are implicitly summed. Here, the term C is the monopole; Ci is a set of three numbers representing the dipole; and so on. In the above expansions, the coefficients may be real or complex. If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties. In the spherical harmonic expansion, we must have In the multi-vector expansion, each coefficient must be real: While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank. This finds use in multipole expansions of the vector potential in electromagnetism, or the metric perturbation in the description of gravitational waves.

For describing functions of three dimensions, away from the coordinate origin, the coefficients of the multipole expansion can be written as functions of the distance to the origin, r— most frequently, as a Laurent series in powers of r. For example, to describe the electromagnetic potential, v , from a source in a small region near the origin, the coefficients may be written as:

Applications of Multipole Expansions Multipole expansions are widely used in problems involving gravitational fields of systems of masses, electric and magnetic fields of charge and current distributions, and the propagation of electromagnetic waves. A classic example is the calculation of the exterior multipole moments of atomic nuclei from their interaction energies with the interior multipoles of the electronic orbitals. The multipole moments of the nuclei report on the distribution of charges within the nucleus and, thus, on the shape of the nucleus. Truncation of the multipole expansion to its first non-zero term is often useful for theoretical calculations. Multipole expansions are also useful in numerical simulations, and form the basis of the Fast Multipole Method of Greengard and Rokhlin, a general technique for efficient computation of energies and forces in systems of interacting particles. The basic idea is to decompose the particles into groups; particles within a group interact normally (i.e., by the full potential), whereas the energies and forces between groups of particles are calculated from their multipole moments. The efficiency of the fast multipole method is generally similar to that of Ewald summation, but is superior if the particles are clustered, i.e., if the system has large density fluctuations. Multipole Expansion of a Potential Outside an Electrostatic Charge Distribution Consider a discrete charge distribution consisting of N point charges qi with position vectors ri. We assume the charges to be clustered around the origin, so that for all i: ri < rmax, where rmax has some finite value. The potential V(R), due to the charge distribution, at a point R outside the charge distribution, i.e., |R| > rmax, can be expanded in powers of 1/R. Two ways of making this expansion can be found in the literature. The first is a Taylor series in the Cartesian coordinates x, y and z, while the second is in terms of spherical harmonics which depend on spherical polar coordinates. The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that the derivations are fairly cumbersome (in fact a large part of it is the implicit rederivation of the Legendre expansion of 1/|r-R|, which was done once and for all by Legendre in the 1780s). Also it is difficult to give a closed expression for a general term of the multipole expansion— usually only the first few terms are given followed by some dots. Expansion in Cartesian Coordinates The Taylor expansion of an arbitrary function v(R-r) around the origin r = 0 is,

with

If v(r-R) satisfies the Laplace equation then the expansion can be rewritten in terms of the components of a traceless Cartesian second rank tensor,

where δαβ is the Kronecker delta and r2 = |r |2. Removing the trace is common, because it takes the rotationally invariant r2 out of the second rank tensor. Example Consider now the following form of v(r-R),

then by direct differentiation it follows that

Define a monopole, dipole and (traceless) quadrupole by, respectively,

and we obtain finally the first few terms of the multipole expansion of the total potential, which is the sum of the Coulomb potentials of the separate charges,

This expansion of the potential of a discrete charge distribution is very similar to the one in real solid harmonics given below. The main difference is that the present one is in terms of linear dependent quantities, for

Note If the charge distribution consists of two charges of opposite sign which are an infinitesimal distance d apart, so that d/R >> (d/ R)2, it is easily shown that the only nonvanishing term in the expansion is

the electric dipolar potential field.

Spherical Form The potential V(R) at a point R outside the charge distribution, i.e., |R| > rmax, can be expanded by the Laplace expansion,

where I;m(R)is an irregular solid harmonic (defined below as a spherical harmonic function divided by Rl+1) and Rm (r) is a regular solid harmonic (a spherical harmonic times rl). We define the spherical multipole moment of the charge distribution as follows

Note that a multipole moment is solely determined by the charge distribution (the positions and magnitudes of the N charges). A spherical harmonic depends on the unit vector Rˆ . (A unit vector is determined by two spherical polar angles.) Thus, by definition, the irregular solid harmonics can be written as

so that the multipole expansion of the field V(R) at the point R outside the charge distribution is given by

This expansion is completely general in that it gives a closed form for all terms, not just for the first few. It shows that the spherical multipole moments appear as coefficients in the 1/R expansion of the potential. It is of interest to consider the first few terms in real form, which are the only terms commonly found in undergraduate textbooks. Since the summand of the m summation is invariant under a unitary transformation of both factors simultaneously and since transformation of complex spherical harmonics to real form is by a unitary transformation, we can simply substitute real irregular solid harmonics and real multipole moments. The l = 0 term becomes

This is in fact Coulomb’s law again. For the l = 1 term we introduce

Then

This term is identical to the one found in Cartesian form. In order to write the l=2 term, we have to introduce shorthand notations for the five real

components of the quadrupole moment and the real spherical harmonics. Notations of the type

can be found in the literature. Clearly the real notation becomes awkward very soon, exhibiting the usefulness of the complex notation. Interaction of Two Non-overlapping Charge Distributions Consider two sets of point charges, one set {qi } clustered around a point A and one set {qj } clustered around a point B. Think for example of two molecules, and recall that a molecule by definition consists of electrons (negative point charges) and nuclei (positive point charges). The total electrostatic interaction energy UAB between the two distributions is

This energy can be expanded in a power series in the inverse distance of A and B. This expansion is known as the multipole expansion of UAB. In order to derive this multipole expansion, we write rXY = rY-rX, which is a vector pointing from X towards Y. Note that We assume that the two distributions do not overlap: Under this condition we may apply the Laplace expansion in the following form

where ILM and RLM are irregular and regular solid harmonics, respectively. The translation of the regular solid harmonic gives a finite expansion,

where the quantity between pointed brackets is a Clebsch-Gordan coefficient. Further we used Use of the definition of spherical multipoles Qml and covering of the summation ranges in a somewhat different order (which is only allowed for an infinite range of L) gives finally

This is the multipole expansion of the interaction energy of two non-overlapping charge distributions which are a distance RAB apart. Since

this expansion is manifestly in powers of 1/RAB. The function Yml is a normalized spherical harmonic. Examples of Multipole Expansions There are many types of multipole moments, since there are many types of potentials and many ways of approximating a potential by a series expansion, depending on the coordinates and the symmetry of the charge distribution. The most common expansions include: • Axial multipole moments of a

potential;

• Spherical multipole moments of a

potential; and

• Cylindrical multipole moments of a ln R potential Examples of potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a potential is the electric ln R potential of an infinite line charge. General Mathematical Properties Mathematically, multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of irreducible representations of the rotational symmetry group, which leads to spherical harmonics and related sets of orthogonal functions. One uses the technique of separation of variables to extract the corresponding solutions for the radial dependencies.

Normal (Geometry) In geometry, an object such as a line or vector is called a normal to another object if they are perpendicular to each other. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

Figure: A polygon and two of its normal vectors

A surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word “normal” is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

The concept has been generalized to differential manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P. In the case of differential curves, the curvature vector is a normal vector of special interest.

Figure: A normal to a surface at a point is the same as a normal to the tangent plane to that surface at that point.

The normal is often used in computer graphics to determine a surface’s orientation toward a light source for flat shading, or the orientation of each of the corners (vertices) to mimic a curved surface with Phong shading. Normal to Surfaces in 3D Space Calculating a Surface Normal: For a convex polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon. For a plane given by the equation ax + by + cz + d = 0, the vector (a,b,c) is a normal. For a plane given by the equation r(α,β) = a + αb + βc, i.e., a is a point on the plane and b and c are (non-parallel) vectors lying on the plane, the normal to the plane is a vector normal to both b and c which can be found as the cross product b × c. For a hyperplane in n+1 dimensions, given by the equation r = a0 + α1a1 + αnan, where a0 is a point on the hyperplane and ai for i = 1, ..., n are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the null space of A where A is given by A = [a1...an]. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. If a (possibly non-flat) surface S is parameterized by a system of curvilinear coordinates x(s, t), with s and t real variables, then a normal is given by the cross product of the partial derivatives If a surface S is given implicitly as the set of points (x,y,z) satisfying F(x,y,z) = 0 , then, a normal at a point (x,y,z) on the surface is given by the gradient

∇F(x, y, z). since the gradient at any point is perpendicular to the level setF(x,y,z) = 0 , and (the surface) is a level set of F. For a surface S given explicitly as a function f(x,y) of the independent variables x,y(e.g., f(x,y) = a00+a01y + a10x + a11xy), its normal can be found in at least two equivalent ways. The first one is obtaining its implicit form F(x,y,z) = z- f(x,y) =0 , from which the normal follows readily as the gradient ∇F(x,y,z). (Notice that the implicit form could be defined alternatively as F(x,y,z) = f(x,y)-z; these two forms correspond to the interpretation of the surface being oriented upwards or downwards, respectively, as a consequence of the difference in the sign of the partial derivative ∂F / ∂z.) The second way of obtaining the normal follows directly from the gradient of the explicit form, ∇f(x,y); by inspection, ∇F(x,y,z) = k-∇f(x,y), where k̂ is the upward unit vector. If a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base. However, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous. Uniqueness of the Normal A normal to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between the inward-pointing normal and outer-pointing normal, which can help define the normal in a unique way. For an oriented surface, the surface normal is usually determined by the right-hand rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

Figure: A vector field of normals to a surface

Transforming Normals

When applying a transform to a surface it is sometimes convenient to derive normals for the resulting surface from the original normals. All points P on tangent plane are transformed to P2 . We want to find n2 perpendicular to P. Let t be a vector on the tangent plane and Ml be the upper 3x3 matrix (translation part of transformation does not apply to normal or tangent vectors).

So use the inverse transpose of the linear transformation (the upper 3x3 matrix) when transforming surface normals. Hypersurfaces in N -dimensional Space The definition of a normal to a surface in three-dimensional space can be extended to (n1) -dimensional hypersurfaces in a n -dimensional space. A hypersurface may be locally defined implicitly as the set of points (x1, x2, ...,xn) satisfying an equation F(x1, x2, ...,xn) = 0, where F is a given scalar function. If F is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not null. At these points the normal vector space has dimension one and is generated by the gradient

The normal line at a point of the hypersurface is defined only if the gradient is not null. It is the line passing through the point and having the gradient as direction. Varieties Defined by Implicit Equations in N -dimensional Space A differential variety defined by implicit equations in the n-dimensional space is the set of the common zeros of a finite set of differential functions in n variables f1(x1, ..., xn),...,fk (x1, ..., xn ). The Jacobian matrix of the variety is the k×n matrix whose i-th row is the gradient of fi. By implicit function theorem, the variety is a manifold in the neighbourhood of a point of it where the Jacobian matrix has rank k. At such a point P, the normal vector space is the vector space generated by the values at P of the gradient vectors of the fi. In other words, a variety is defined as the intersection of k hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. The normal (affine) space at a point P of the variety is the affine subspace passing through P and generated by the normal vector space at P.

These definitions may be extended verbatim to the points where the variety is not a manifold. Example Let V be the variety defined in the 3-dimensional space by the equations xy = 0, z= 0. This variety is the union of the x-axis and the y-axis. At a point (a, 0, 0) where a≠0, the rows of the Jacobian matrix are (0, 0, 1) and (0, a, 0). Thus the normal affine space is the plane of equation x=a. Similarly, if b≠0, the normal plane at (0, b, 0) is the plane of equation y=b. At th e p oin t (0, 0, 0) t h e r ow s of th e Jac obian matr ix ar e (0, 0 , 1 ) and (0,0,0). Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z-axis. Uses • Surface normals are essential in defining surface integrals of vector fields. • Surface normals are commonly used in 3D computer graphics for lighting calculations. • Surface normals are often adjusted in 3D computer graphics by normal mapping. • Render layers containing surface normal information may be used in Digital compositing to change the apparent lighting of rendered elements. Normal in Geometric Optics The normal is the line perpendicular to the surface of an optical medium. In reflection of light, the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray and the angle between the normal and the reflected ray.

Figure: Diagram of specular reflection

Parallelogram of Force The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object.

Figure: Parallelogram construction for adding vectors

When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. Forces, being vectors are observed to obey the laws of vector addition, and so the overall (resultant) force due to the application of a number of forces can be found geometrically by drawing vector arrows for each force. This construction has the same result as moving F2 so its tail coincides with the head of F1, and taking the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated to add F3 to the resultant F1 + F2, and so forth. Proof

Figure: Parallelogram of velocity

Preliminary: the Parallelogram of Velocity Suppose a particle moves at a uniform rate along a line from A to B in a given time (say, one second), while in the same time, the line AB moves uniformly from its position at AB to a position at DC, remaining parallel to its original orientation throughout. Accounting for both motions, the particle traces the line AC. Because a displacement in a given time is a measure of velocity, the length of AB is a measure of the particle’s velocity along AB, the length of AD is a measure of the line’s velocity along AD, and the length of AC is a measure of the particle’s velocity along AC. The particle’s motion is the same as if it had moved with a single velocity along AC. Newton’s Proof of the Parallelogram of Force Suppose two forces act on a particle at the origin (the “tails” of the vectors) of Figure given above. Let the lengths of the vectors F1 and F2 represent the velocities the two forces could produce in the particle by acting for a given time, and let the direction of each represent the direction in which they act. Each force acts independently and will produce its particular velocity whether the other force acts or not. At the end of the given time, the particle has both velocities. By the above proof, they are equivalent to a single velocity, Fnet. By Newton’s second law, this vector is also a measure of the force which would produce that velocity, thus the two forces are equivalent to a single

force.

Figure: Using a parallelogram to add the forces acting on a particle on a smooth slope. We find, as we’d expect, that the resultant (double headed arrow) force acts down the slope, which will cause the particle to accelerate in that direction.

Controversy The proof of the parallelogram of force was not generally accepted without some controversy. Various proofs were developed (chiefly Duchayla’s and Poisson’s), and these also caused objections. That the parallelogram of force was true was not questioned; why it was true was. This continued throughout the 19th century and into the early 20th.

Poloidal Toroidal Decomposition In vector analysis, a mathematical discipline, a three-dimensional solenoidal vector field F can be considered to be generated by a pair of scalar potentials *F and O: where k is a unit vector. This decomposition is analogous to a Helmholtz decomposition, and has been used in dynamo theory.

Potential Gradient In physics, chemistry and biology, a potential gradient is the local rate of change of the potential with respect to displacement, i.e. spatial derivative, or gradient. This quantity frequently occurs in equations of physical processes because it leads to some form of flux. In electrical engineering it refers specifically to electric potential gradient, which is equal to the electric field. Definition Elementary Algebra/calculus: Fundamentally - the expression for a potential gradient F in one dimension takes the form: where Φ is some type of potential, and x is displacement (not distance), in the x direction. In the limit of infinitesimal displacements, the ratio of differences becomes a ratio of differentials:

In three dimensions, the resultant potential gradient is the sum of the potential gradients in each direction, in Cartesian coordinates:

where ex, ey, ez are unit vectors in the x, y, z directions, which can be compactly and neatly written in terms of the gradient operator ∇, Vector Calculus The mathematical nature of a potential gradient arises from vector calculus, which directly has application to physical flows, fluxes and gradients over space. For any conservative vector field F, there exists a scalar field 0, such that the gradient ∇ of the scalar field is equal to the vector field; using Stoke’s theorem, this is equivalently stated as ∇×F=0 meaning the curl ∇ × of the vector field vanishes. In physical problems, the scalar field is the potential, and the vector field is a force field, or flux/current density describing the flow some property. Physics In the case of the gravitational field g, which can be shown to be conservative, it is equal to the gradient in gravitational potential Φ: Notice the opposite signs between gravitational field and potential - because as the potential gradient and field are opposite in direction, as the potential gradient increases, the gravitational field strength decreases and vice versa. In electrostatics, the electrostatic (not dynamic) field has identical properties to the gravitational field; it is the gradient of the electric potential -E = ∇V The electrodynamic field has a non-zero curl, which implies the electric field cannot be only equal to the gradient in electric potential, a time-dependent term must be added;

where A is the electromagnetic vector potential. Chemistry In an Electrochemical half-cell, at the interface between the electrolyte (an ionic solution) and the metal electrode, the standard electric potential difference is;

where R =gas constant, T =temperature of solution, z =valency of the metal, e = elementary charge, NA = Avagadro’s constant, and aM+z is the activity of the ions in solution. Quantities with superscript o denote the measurement is taken under standard conditions. The potential gradient is relatively abrupt, since there is an almost definite boundary between the metal and

solution, hence the term interface.

Pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true or polar vector (more formally, a contravariant vector), which on reflection matches its mirror image.

Figure: A loop of wire (black), carrying a current, creates a magnetic field (blue). If the position and current of the wire are reflected across the dotted line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position of the wire and its current are vectors, but the magnetic field is a pseudovector.

In three dimensions the pseudovector p is associated with the cross product of two polar vectors a and b: p = a × b. The vector p calculated this way is a pseudovector. One example is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, which can be said to span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics where it has to be considered when transforming surface normals. A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics pseudovectors are equivalent to three dimensional bivectors, from which the transformation rules of pseudovectors can be derived. More generally in n-dimensional geometric algebra pseudovectors are the elements of the algebra with dimension n - 1, written av_1Rn. The label ‘pseudo’ can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor. Physical Examples Physical examples of pseudovectors include the magnetic field, torque, vorticity, and the angular momentum. Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems. For example, consider the case of an electrical current loop in the z = 0 plane, which has a magnetic field at z = 0 that is oriented in the z direction. This system is symmetric (invariant) under mirror reflections through the plane (an

improper rotation), so the magnetic field should be unchanged by the reflection. But reflecting the magnetic field through that plane naively appears to change its sign if it is viewed as a vector field—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature, so the mirror flip in the end leaves the magnetic field unchanged as expected. As another example, consider the pseudovector angular momentum L = r × p. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the “reflection” of this angular momentum “vector” (viewed as an ordinary vector) points to the right, but the actual angular momentum vector of the wheel still points to the left, corresponding to the extra minus sign in the reflection of a pseudovector. This reflects the fact that the wheels are still turning forward. In comparison, the behaviour of a regular vector, such as the position of the car, is quite different.

Figure: Each wheel of a car driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car.

To the extent that physical laws would be the same if the universe were reflected in a mirror (equivalently, invariant under parity), the sum of a vector and a pseudovector is not meaningful. However, the weak force, which governs beta decay, does depend on the chirality of the universe, and in this case pseudovectors and vectors are added. Details The definition of a “vector” in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of “vector” (namely, any element of an abstract vector space). Under the physics definition, a “vector” is required to have components that “transform” in a certain way under a proper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of active transformations.) Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector × is transformed to x2 = Rx, then any “vector” v must be similarly transformed to v2 = Rv. This important requirement is what distinguishes a vector (which might be composed of, for example, the x, y, and z-components of velocity) from any other triplet of physical quantities (For ex ample, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.) (In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one.)

The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider improper rotations, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is inversion.) Suppose everything in the universe undergoes an improper rotation described by the rotation matrix R, so that a position vector × is transformed to x2 = Rx. If the vector v is a polar vector, it will be transformed to v2 = Rv. If it is a pseudovector, it will be transformed to v2 = -Rv. The transformation rules for polar vectors and pseudovectors can be compactly stated as v′ = Rv (polar vector) v′ = (det R)(Rv) (pseudovector) where the symbols are as described above, and the rotation matrix R can be either proper or improper. The symbol det denotes determinant; this formula works because the determinant of proper and improper rotation matrices are +1 and -1, respectively. Behaviour Under Addition, Subtraction, Scalar Multiplication Suppose v1 and v2 are known pseudovectors, and v3 is defined to be their sum, v3=v1+v2. If the universe is transformed by a rotation matrix R, then v3 is transformed to So v3 is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector. On the other hand, suppose v1 is known to be a polar vector, v2 is known to be a pseudovector, and v3 is defined to be their sum, v3=v1+v2. If the universe is transformed by a rotation matrix R, then v3 is transformed to Therefore, v3 is neither a polar vector nor a pseudovector. For an improper rotation, v3 does not in general even keep the same magnitude: If the magnitude of v3 were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the weak interaction: Certain radioactive decays treat “left” and “right” differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. Behaviour Under Cross Products For a rotation matrix R, either proper or improper, the following mathematical equation is always true: where v1 and v2 are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation, and is well known.)

Figure: Under inversion the two vectors change sign, but their cross product is invariant [black are the two original vectors, grey are the inverted vectors, and red is their mutual cross product].

Suppose v1 and v2 are known polar vectors, and v3 is defined to be their cross product, v3=v1×v2. If the universe is transformed by a rotation matrix R, then v3 is transformed to So v3 is a pseudovector. Similarly, one can show: • • • •

polar vector × polar vector = pseudovector pseudovector × pseudovector = pseudovector polar vector × pseudovector = polar vector pseudovector × polar vector = polar vector

Examples From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Continuing this way, it is straightforward to classify any vector as either a pseudovector or polar vector. The Right-hand Rule Above, pseudovectors have been discussed using active transformations. An alternate approach, more along the lines of passive transformations, is to keep the universe fixed, but switch “right-hand rule” with “left-hand rule” and vice-versa everywhere in physics, in particular in the definition of the cross product. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in the parity-violating phenomena such as certain radioactive decays. Geometric Algebra In geometric algebra the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors. The basic multiplication in the geometric algebra is the geometric product, denoted by simply juxtaposing two vectors as in ab. This product is expressed as: ab = a-b + a ∧ b, where the leading term is the customary vector dot product and the second term is called the

wedge product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A kfold wedge product also is referred to as a k-blade. In the present context the pseudovector is one of these combinations. This term is attached to a different multivector depending upon the dimensions of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2blade or bivector can be expressed as a single wedge product and is a pseudovector. In four dimensions, however, the pseudovectors are trivectors. In general, it is a (n - 1)blade, where n is the dimension of the space and algebra. An n-dimensional space has n vectors and also n pseudovectors. Each pseudovector is formed from the outer (wedge) product of all but one of the n vectors. For instance, in four dimensions where the vectors are: {e1, e2, e3, e4}, the pseudovectors can be written as: {e234, e134, e124, e123} Transformations in Three Dimensions The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis. He says: “The terms axial vector and pseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual.” To paraphrase Baylis: Given two polar vectors (that is, true vectors) a and b in three dimensions, the cross product composed from a and b is the vector normal to their plane given by c = a × b. Given a set of right-handed orthonormal basis vectors { e £ }, the cross product is expressed in terms of its components as: where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ≠ b. In this context of geometric algebra, this bivector is called a pseudovector, and is the dual of the cross product. The dual of e1 is introduced as e23 a” e2e3 = e2 ≠ e3, and so forth. That is, the dual of e1 is the subspace perpendicular to e1, namely the subspace spanned by e2 and e3. With this understanding, Comparison shows that the cross product and wedge product are related by: a ∧ b = i a × b, where i = e1 ≠ e2 ≠ e3 is called the unit pseudoscalar. It has the property: i2 = -1. Using the above relations, it is seen that if the vectors a and b are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors e–! are inverted, then the pseudovector is invariant, but the cross product changes sign. This behaviour of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

Note on Usage As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product. However, because the cross product does not generalize beyond three dimensions, the notion of pseudovector based upon the cross product also cannot be extended to higher dimensions. The pseudovector as the (n–1)-blade of an n-dimensional space is not so restricted. Another important note is that pseudovectors, despite their name, are “vectors” in the common mathematical sense, i.e. elements of a vector space. The idea that “a pseudovector is different from a vector” is only true with a different and more specific definition of the term “vector” as discussed above.

Scalar Potential A scalar potential is a fundamental concept in vector analysis and physics (the adjective scalar is frequently omitted if there is no danger of confusion with vector potential). The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that:

where VP is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x,y,z. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length. In order for F to be described in terms of a scalar potential only, the following have to be true: 1.

where the integration is over a Jordan arc passing from location a to location b and P(b) is P evaluated at location b .

2.

where the integral is over any simple closed path, otherwise known as a Jordan curve.

3. ∇ × F = 0 The first of these conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar field P. The second condition is a requirement of F so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the curl of F using the fundamental theorem of the curl. A vector field F that satisfies these conditions is said to be irrotational (Conservative). Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar potential associated with the gravity per unit mass, i.e., the acceleration due to the field, as a function of position. The gravity potential is the gravitational potential energy per unit mass. In electrostatics the electric potential is the

scalar potential associated with the electric field, i.e., with the electrostatic force per unit charge. The electric potential is in this case the electrostatic potential energy per unit charge. In fluid dynamics, irrotational lamellar fields have a scalar potential only in the special case when it is a Laplacian field. Certain aspects of the nuclear force can be described by a Yukawa potential. The potential play a prominent role in the Lagrangian and Hamiltonian formulations of classical mechanics. Further, the scalar potential is the fundamental quantity in quantum mechanics. Not every vector field has a scalar potential. Those that do are called conservative, corresponding to the notion of conservative force in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential. In electrodynamics the electromagnetic scalar and vector potentials are known together as the electromagnetic four-potential. Integrability Conditions If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference point r0 is defined in terms of the line integral: where C is a parametrized path from to The fact that the line integral depends on the path C only through its terminal points and is, in essence, the path independence property of a conservative vector field. The fundamental theorem of calculus for line integrals implies that if V is defined in this way, then F =-VV, so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point r0. Altitude as Gravitational Potential Energy An example is the (nearly) uniform gravitational field near the Earth’s surface. It has a potential energy U = mgh where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the threedimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill’s surface, which cancels out the component of gravity perpendicular to the hill’s surface. The component of gravity that remains to move the ball is parallel to the

surface: FS = -mg sin θ where θ is the angle of inclination, and the component of FS perpendicular to gravity is

Figure: uniform gravitational field near the Earth’s surface

Figure: Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.

This force FP, parallel to the ground, is greatest when θ is 45 degrees. Let Δh be the uniform interval of altitude between contours on the contour map, and let Δx be the distance between two contours. Then

so that

However, on a contour map, the gradient is inversely proportional to Δx, which is not similar to force FP: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth’s surface represented by the contour map. Pressure as Buoyant Potential In fluid mechanics, a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This buoyant force is the negative gradient of pressure:

fB = -∇p. Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the ground. The surface of the water can be characterized as a plane with zero pressure. If the liquid has a vertical vortex (whose axis of rotation is perpendicular to the ground), then the vortex causes a depression in the pressure field. The surfaces of constant pressure are parallel to the ground far away from the vortex, but near and inside the vortex the surfaces of constant pressure are pulled downwards, closer to the ground. This also happens to the surface of zero pressure. Therefore, inside the vortex, the top surface of the liquid is pulled downwards into a depression, or even into a tube (a solenoid). The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative pressure gradient along the surface of the object: A moving airplane wing makes the air pressure above it decrease relative to the air pressure below it. This creates enough buoyant force to counteract gravity. Calculating the Scalar Potential Given a vector field E, its scalar potential cPcan be calculated to be

where r is volume. Then, if E is irrotational (Conservative),

This formula is known to be correct if E is continuous and vanishes asymptotically to zero towards infinity, decaying faster than 1/r and if the divergence of E likewise vanishes towards infinity, decaying faster than 1/r2.

Skew Gradient In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function, and has the same magnitude that the gradient has. Definition The skew gradient can be defined using complex analysis and the Cauchy–Riemann equations. Let f(z(x, y)) = u(x, y) + iv(x, y) be a complex-valued analytic function, where u, v are real-valued scalar functions of the real variables x, y. A skew gradient is defined as: and from the Cauchy-Riemann equations, it is derived that

Properties The skew gradient has two interesting properties. It is everywhere orthogonal to the gradient of u, and of the same length:

Chapter 5: Solenoidal Vector Field In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero at all points in the field:

Properties The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: automatically results in the identity (as can be shown, for example, using Cartesian coordinates): The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ × A. (Strictly speaking, this holds only subject to certain technical conditions on v.) The divergence theorem, gives the equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where ds is the outward normal to each surface element. Etymology Solenoidal has its origin in the Greek word for solenoid, which is meaning pipe-shaped, from pipe. In the present context of solenoidal it means constrained as if in a pipe, so with a fixed volume. Examples • • • •

The magnetic flux density B is solenoidal; The velocity field of an incompressible fluid flow is solenoidal; The vorticity field is solenoidal The electric flux density D in regions where there is no charge (ρe = 0);

• The current density J where the charge density is unvarying,

.

Surface Gradient In vector calculus, the surface gradient is a vector differential operator that is similar to the conventional gradient. The distinction is that the surface gradient takes effect along a surface. For a surface S in a scalar field u, the surface gradient is defined and notated as

where n̂ is a unit normal to the surface. Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the surface removed (subtracted), hence this gradient is tangent to the surface. In other words, the surface gradient is the orthographic projection of the gradient onto the surface. The surface gradient arises whenever the gradient of a quantity over a surface is important. In the study of capillary surfaces for example, the gradient of spatially varying surface tension doesn’t make much sense, however the surface gradient does and serves certain purposes.

Triangulation in Three Dimensions Triangulation in three dimensions is a method of finding the location of a point in three dimensions based on other known coordinates and distances, it is commonly used in surveying and astronomy. Triangulation is also used in 2 dimensions to find the location of a point on a plane, this is commonly used in navigation to plot positions on a map. One Method to Triangulate a Location in 3D This method uses vector analysis to determine the coordinates of the point where three lines meet given the scalar lengths of the lines and the coordinates of their bases. First treat these three lines as if they are the radii of three spheres of known centres (these known centres being the coordinates of the known end of each line), this method can then be used to calculate the intersection of the three spheres if they intersect. In the event that the three spheres don’t intersect, this method obtains the closest solution to the axis of symmetry between three spheres. Development Three sticks of known lengths AD, BD, CD are anchored in the ground at known coordinates A, B, C. This development calculates the coordinates of the apex where the other ends of the three sticks will meet. These coordinates are given by the vector D. In the mirror case, D’ is sub-apex where the three sticks would meet below the plane of A, B, C as well.

Figure: Apex and its mirror reflection about the plane of ABC precipitate D and D’.

By the law of cosines,

Figure: The normals are dropped on the sides from the apex and their intersections with AB, AC and BC are determined.

The projection of AD onto AB and AC, and the projection of BD onto BC results in,

Figure: The red normals intersect at a common point.

The three unit normals to AB, AC and BC in the plane ABC are:

Then the three vectors intersect at a common point: Solving for mAB, mAC and mBC

Spreadsheet Formula A spreadsheet command for calculating this is, Product(product(minverse(product(transpose H, H)), Transpose H), G)

An example of a spreadsheet that does complete calculations of this entire problem is given at the External links section at the end of this article. The matrix H and the matrix g in this least squares solution are,

Alternatively, solve the system of equations for mAB, mAC and mBC:

The unit normal to the plane of ABC is,

Solution

where

Condition for Intersection If AD, BD, CD are assigned according to the arrangement, AD ≤ BD ≤ CD Then AD, BD, CD intersect if and only if,

Viz, if

such that, then the three spheres intersect if and only if,

Decoding Vector Formulas

The equation of the line of the axis of symmetery of 3 spheres

Vector Field In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).

Figure: A portion of the vector field (sin y, sin x)

In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field. Definition Vector fields on subsets of Euclidean space:

Figure: Two representations of the same vector field: v(x, y) = -r . The arrows depict the field at discrete points, however, the field exists everywhere.

Given a subset S in Rn, a vector field is represented by a vector-valued function V:S→Rnin standard Cartesian coordinates (x1, ..., xn ). If each component of V is continuous, then V is a continuous vector field, and more generally V is a Ck vector field if each component V is k times continuously differ entiable. A vector field can be visualized as assigning a vector to individual points within an ndimensional space. Given two Ck-vector fields V, W defined on S and a real valued Ck-function f defined on S, the two operations scalar multiplication and vector addition

define the module of Ck-vector fields over the ring of Ck-functions. Coordinate Transformation Law In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector. Thus, suppose that (x1,...,xn) is a choice of Cartesian coordinates, in terms of which the coordinates of the vector V are Vx = (V1,x ,…,Vn,x ) and suppose that (y1,...,yn) are n functions of the xi defining a different coordinate system. Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law

Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law (1) relating the different coordinate systems. Vector fields are thus contrasted with scalar fields, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes. Vector Fields on Manifolds

Figure: A vector field on a sphere

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that p°Fis the identity mapping where p denotes the projection from TM to M. In other words, a vector field is a section of the tangent bundle. If the manifold M is smooth (respectively analytic)—that is, the change of coordinates are smooth (respectively analytic)— then one can make sense of the notion of smooth (respectively analytic) vector fields. The collection of all smooth vector fields on a smooth manifold M is often denoted by Γ(TM) or C”(M,TM) (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by X(M) (a fraktur “X”). Examples

Figure: The flow field around an airplane is a vector field in R3 , here visualized by bubbles that follow the streamlines showing a wingtip vortex.

• A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A “high” on the usual barometric pressure map would then act as a source (arrows pointing away), and a “low” would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas. • Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid. • Streamlines, Streaklines and Pathlines are 3 types of lines that can be made from vector fields. They are : o streaklines — as revealed in wind tunnels using smoke. o streamlines (or fieldlines)— as a line depicting the instantaneous field at a given

time. o pathlines — showing the path that a given particle (of zero mass) would follow. • Magnetic fields. The fieldlines can be revealed using small iron filings. • Maxwell’s equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field. • A gravitational field generated by any massive object is also a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere’s centre with the magnitude of the vectors reducing as radial distance from the body increases. Gradient Field

Figure: A vector field that has circulation about a point cannot be written as the gradient of a function.

Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇ ). A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that

The associated flow is called the gradient flow, and is used in the method of gradient descent. The path integral along any closed curve y (X0) = X1)) in a gradient field is zero:

Central Field A C-vector field over Rn \ {0} is called a central field if where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called the centre of the field. Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since

defining it on one semiaxis and integrating gives an antigradient. Operations on Vector Fields Line Integral: A common technique in physics is to integrate a vector field along a curve, i.e. to determine its line integral. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at a given point in space, the line integral is the work done on the particle when it travels along a certain path. The line integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector field V and a curve y parametrized by [0, 1] the line integral is defined as

Divergence The divergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by

with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem. The divergence can also be defined on a Riemannian manifold, that is, a manifold with a Riemannian metric that measures the length of vectors. Curl The curl is an operation which takes a vector field and produces another vector field. The curl is defined only in three-dimensions, but some properties of the curl can be captured in higher dimensions with the exterior derivative. In three-dimensions, it is defined by

The curl measures the density of the angular momentum of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes’ theorem.

Chapter 6: Coordinate System In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in ‘the x-coordinate’. In elementary mathematics the coordinates are taken to be real numbers, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry. An example in everyday use is the system of assigning longitude and latitude to geographical locations. In physics, a coordinate system used to describe points in space is called a frame of reference.

Number Line In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from -9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. It is divided into two symmetric halves by the origin, i.e. the number zero. In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa. Drawing the Number Line The number line is usually represented as being horizontal. Customarily, positive numbers lie on the right side of zero, and negative numbers lie on the left side of zero. An arrowhead on either end of the drawing is meant to suggest that the line continues indefinitely in the positive and negative real numbers, denoted by ℝ. The real numbers consist of irrational numbers and rational numbers, as well as the integers, whole numbers, and the natural numbers (the counting numbers). A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.

Cartesian Coordinate System

Figure: Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in green, (-3, 1) in red, (-1.5, -2.5) in blue, and the origin (0, 0) in purple.

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0,0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, one can specify a point in a space of any dimension n by use of n Cartesian coor dinates, the sig ned distan ces fr om n mutually perpendicular hyperplanes.

Figure: Cartesian coordinate system with a circle of radius 2 centreed at the origin marked in red. The equation of a circle is (x – a)2 + (y – b)2 = r2 where a and b are the coordinates of the centre (a, b) and r is the radius.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4. Cartesian coordinates are the foundation of analytic geometry, and provide enlightening

geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data processing. History The adjective Cartesian refers to the French mathematician and philosopher René Descartes (who used the name Cartesius in Latin). The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery. Both authors used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced in later work by commentators who were trying to clarify the ideas contained in Descartes’ La Géométrie. The development of the Cartesian coordinate system would play an intrinsic role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. Nicole Oresme, a French cleric and friend of the dauphin (later to become King Charles V) of the 14th Century, used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

Definitions Number Line Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—means choosing a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative; we then say that the line “is oriented” (or “points”) from the negative half towards the positive half. Then each point p of the line can be specified by its distance from O, taken with a + or – sign depending on which half-line contains p. A line with a chosen Cartesian system is called a number line. Every real number, whether integer, rational, or irrational, has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers. Cartesian Coordinates in Two Dimensions The modern Cartesian coordinate system in two dimensions (also called a rectangular

coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. (Early systems allowed “oblique” axes, that is, axes that did not meet at right angles.) The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For a given point P, a line is drawn through P perpendicular to the x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding number lines. The coordinates are written as an ordered pair (x, y). The point where the axes meet is the common origin of the two number lines and is simply called the origin. It is often labeled O and if so then the axes are called Ox and Oy. A plane with x and y-axes defined is often referred to as the Cartesian plane or xy plane. The value of x is called the x-coordinate or abscissa and the value of y is called the ycoordinate or ordinate. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. In the Cartesian plane, reference is sometimes made to a unit circle or a unit hyperbola. Cartesian Coordinates in Three Dimensions

Figure: A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates × = 2, Y = 3, and Z = 4, or (2,3,4).

Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines. Alternatively, the coordinates of a point p can also be taken as the (signed) distances from p to the three planes defined by the three axes. If the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes. The distance is to be taken with the + or – sign, depending on which of the two half-spaces separated by that plane contains p. The y and z coordinates can be obtained in the same way from the (x, z) and (x, y)

planes, respectively.

Figure: The coordinate surfaces of the Cartesian coordinates (x, y, z). The z-axis is vertical and the x-axis is highlighted in green. Thus, the red plane shows the points with x= 1, the blue plane shows the points with z= 1, and the yellow plane shows the points with y= -1. The three surfaces intersect at the point P (shown as a black sphere) with the Cartesian coordinates (1, -1, 1).

Generalizations One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and/or different units along each axis. In that case, each coordinate is obtained by projecting the point onto one axis along a direction that is parallel to the other axis (or, in general, to the hyperplane defined by all the other axes). In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas (such as the Pythagorean formula for the distance) do not hold. Notations and Conventions The Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10,5) or (3,5,7). The origin is often labelled with the capital letter O. In analytic geometry, unknown or generic coordinates are often denoted by the letters x and y on the plane, and x, y, and z in three-dimensional space. w is often used for four-dimensional space, but the rarity of such usage precludes concrete convention here. This custom comes from an old convention of algebra, to use letters near the end of the alphabet for unknown values (such as were the coordinates of points in many geometric problems), and letters near the beginning for given quantities. These conventional names are often used in other domains, such as physics and engineering. However, other letters may be used too. For example, in a graph showing how a pressure varies with time, the graph coordinates may be denoted t and P. Each axis is usually named after the coordinate which is measured along it; so one says the x-axis, the y-axis, the t-axis, etc. Another common convention for coordinate naming is to use subscripts, as in x1, x2, ... xn for the n coordinates in an n-dimensional space; especially when n is greater than 3, or variable. Some authors (and many programmers) prefer the numbering x0, x1, ... xn-1. These notations are especially advantageous in computer programming: by storing the coordinates of a point as an array, instead of a record, one can use iterative commands or procedure parametres instead of repeating the same commands for each coordinate. In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called

the abscissa) is measured along a horizontal axis, oriented from left to right. The second coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top. However, in computer graphics and image processing one often uses a coordinate system with the y axis pointing down (as displayed on the computer’s screen). This convention developed in the 1960s (or earlier) from the way that images were originally stored in display buffers. For three-dimensional sy st em s, t h e z axis is often shown vertical an d poin tin g up (positiv e up), so that the x and y axes lie on a horizontal plane. If a diagram (3D projection or 2D perspective drawing) shows the x and y axis horizontally and vertically, respectively, then the z axis should be shown pointing “out of the page” towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise. All laws of physics and math assume this right-handedness, which ensures consistency. For 3D diagrams, the names “abscissa” and “ordinate” are rarely used for x and y, respectively. When they are, the zcoordinate is sometimes called the applicate. The words abscissa, ordinate and applicate are sometimes used to refer to coordinate axes rather than values.

Octant (Solid Geometry) An octant in solid geometry is one of the eight divisions of a Euclidean threedimensional coordinate system defined by the signs of the coordinates. It is similar to the two-dimensional quadrant and the one-dimensional ray. The generalization of an octant is called orthant. Numbering

Figure: For z > 0, the octants have the same numbers as the corresponding quadrants in the plane.

Usually, the octant with all three positive coordinates is referred to as the first octant. There is no generally used naming convention for the other seven octants.

Quadrant (Plane Geometry)

Figure: The four quadrants of a Cartesian coordinate system.

The axes of a two-dimensional Cartesian system divide the plan e in to fou r in finite reg ion s, called qu adr ants, eac h bou n ded b y t w o half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (-,+), III (-,–), and IV (+,-). When the axes are drawn according to the mathematical custom, the numbering goes c oun ter -clockwise starting from the upper right (“northeast”) quadrant. Cartesian Space A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are unique and non-ambiguous, the points of a plane can be identified with all possible pairs of real numbers; that is with the product R2 =1×1, where ℝ is the set of all reals. In the same way one defines a space of any dimension n, whose points can be identified with the tuples (lists) numbers, that is, with W .

Cartesian Cartesian Cartesian of n real

Cartesian Formulas for the Plane Distance Between Two Points The Euclidean distance between two points of the plane with Cartesian coordinates (x1, y1)and (x2,y2)is

This is the Cartesian version of Pythagoras’ theorem. In three-dimensional space, the distance between points which can be obtained by two consecutive applications of Pythagoras’ theorem. Euclidean Transformations The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points. There are four types of these mappings (also called isometries): translations, rotations, reflections and glide reflections. Translation Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (a,b) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are (x,y), after the translation they will be (x′,y′) = (x + a,y + b). Rotation To rotate a figure counterclockwise around the origin by some angle θ is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x’,y), where

Reflection If (x, y) are the Cartesian coordinates of a point, then (-x, y) are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror. Likewise, (x, -y) are the coordinates of its reflection across the first coordinate axis (the × axis). In more generality, reflection across a line through the origin making an angle θwith the x-axis, is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x’,y), where

Glide Reflection A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection). General Matrix Form of the Transformations These Euclidean transformations of the plane can all be described in a uniform way by using matrices. The result (x′, y′) of applying a Euclidean transformation to a point (x, y) is

given by the formula where A is a 2×2 orthogonal matrix and b = (b1, b2) is an arbitrary ordered pair of numbers; that is,

where . [Note the use of row vectors for point coordinates and that the matrix is written on the right.] To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is, and This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero. The formula defines a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that A reflection or glide reflection is obtained when, Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices. Affine Transformation Another way to represent coordinate transformations in Cartesian coordinates is through affine transformations. In affine transformations an extra dimension is added and all points are given a value of 1 for this extra dimension. The advantage of doing this is that then all of the euclidean transformations become linear transformations and can be represented using matrix multiplication. The affine transformation is given by:

[Note the A matrix from above was transposed. The matrix is on the left and column vectors for point coordinates are used.] Using affine transformations multiple different euclidean transfor-mations including translation can be combined by simply multiplying the corresponding matrices. Scaling An example of an affine transformation which is not a Euclidean motion is given by scaling.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If (x,y) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates (x′,y′) = (mx,my). If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller. Shearing A shearing transformation will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by: (x′,y′) = (x + ys,y) Shearing can also be applied vertically: (x′,y′) = (x,xs + y)

Orientation (Vector Space)

Figure: The left-handed orientation is shown on the left, and the right-handed on the right.

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions. In this setting, the orientation of an ordered basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a rotation alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral). The orientation on a real vector space is the arbitrary choice of which ordered bases are “positively” oriented and which are “negatively” oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation is called an oriented vector space, while one without a choice of orientation is called unoriented. Definition

Let V be a finite-dimensional real vector space and let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class and -1 to the other. Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on Rn provides a standard orientation on Rn (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between V and Rn will then provide an orientation on V. The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a nonsingular linear mapping of vector space Rn to Rn. This mapping is orientation-preserving if its determinant is positive. For instance, in R3 a rotation around the Z Cartesian axis by an angle σ is orientation-preserving:

while a reflection by the XY Cartesian plane is not orientation-preserving:

Zero-dimensional Case The concept of orientation defined above did not quite apply to zero-dimensional vector spaces (as the only empty matrix is the identity (with determinant 1), so there will be only one equivalence class). However, it is useful to be able to assign different orientations to a point (e.g. orienting the boundary of a 1-dimensional manifold). A more general definition of orientation that works regardless of dimension is the following: An orientation on V is a map from the set of ordered bases of V to the set {+1}that is invariant under base changes with positive determinant and changes sign under base changes with negative determinant (it is equivarient with respect to the homomorphism GLn → ±1). The set of ordered bases of the zero-dimensional vector space has one element (the empty set), and so there are two maps from this set to{±l}. A subtle point is that a zero-dimensional vector space is naturally (canonically) oriented,

so we can talk about an orientation being positive (agreeing with the canonical orientation) or negative (disagreeing). An application is interpreting the Fundamental theorem of calculus as a special case of Stokes’ theorem. Two ways of seeing this are: • A zero-dimensional vector space is a point, and there is a unique map from a point to a point, so every zero-dimensional vector space is naturally identified with R0, and thus is oriented. • The 0th exterior power of a vector space is the ground field, which here is R1, which has an orientation (given by the standard basis).

Alternate Viewpoints Multilinear Algebra For any n-dimensional real vector space V we can form the kth-exterior power of V, denoted ∇k V. This is a real vector space of dimension . The vector space ∇nV (called the top exterior power) therefore has dimension 1. That is, ∇nV is just a real line. There is no a priori choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero linear form co on ËnV determines an orientation of V by declaring that x is in the positive direction when co(x) > 0. To connect with the basis point of view we say that the positively oriented bases are those on which co evaluates to a positive number (since co is an n-form we can evaluate it on an ordered set of n vectors, giving an element of R). The form co is called an orientation form. If {ei} is a privileged basis for V and {e*} is the dual basis, then the orientation form giving the standard orientation is e1* ∇ e2* ∇...∇en*. The connection of this with the determinant point of view is: the determinant of an endomorphism T : V → V can be interpreted as the induced action on the top exterior power.

Orientation on Manifolds One can also discuss orientation on manifolds. Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. One can assign to each of these vector spaces an orientation. However, one would like to know whether it is possible to choose the orientations so that they “vary smoothly” from point to point. Due to certain topological restrictions, there are situations when this is impossible. A manifold which admits a smooth choice of orientations for its tangents spaces is said to be orientable.

Axes Conventions

Figure: Heading, elevation and bank angles (Z-Y’-X’’) for an aircraft. The aircraft’s pitch and yaw axes Y and Z are not shown, and its fixed reference frame xyz has been shifted backwards from its centre of gravity (preserving angles) for clarity. Axes named according to the air norm DIN 9300

Mobile objects are normally tracked from an external frame considered fixed. Other frames can be defined on those mobile objects to deal with relative positions for other objects. Finally, attitudes or orientations can be described by a relationship between the external frame and the one defined over the mobile object. The orientation of a vehicle is normally referred to as attitude. It is described normally by the orientation of a frame fixed in the body relative to a fixed reference frame. The attitude is described by attitude coordinates, and consists of at least three coordinates. While from a geometrical point of view the different methods to describe orientations are defined using only some reference frames, in engineering applications it is important also to describe how these frames are attached to the lab and the body in motion. Due to the special importance of international conventions in air vehicles, several organizations have published standards to be followed. For example, German DIN has published the DIN 9300 norm for aircraft (adopted by ISO as ISO 1151– 2:1985). Ground Reference Frames: ENU and NED Basically, as lab frame or reference frame, there are two kinds of conventions for the frames (sometimes named LVLH, local vertical, local horizontal): • East, North, Up, referred as ENU • North, East, Down, referred as NED, used specially in aerospace These frames are location dependent. For movements around the globe, like air or sea navigation, the frames are defined as tangent to the lines of coordinates. • East-West tangent to parallels, • North-South tangent to meridians, and • Up-Down in the direction to the centre of the earth (when using a spherical Earth simplification), or in the direction normal to the local tangent plane (using an oblate spheroidal or geodetic ellipsoidal model of the earth) which does not generally pass through the centre of the Earth. To establish a standard convention to describe attitudes, it is required to establish at least the axes of the reference system and the axes of the rigid body or vehicle. When an ambiguous notation system is used (such as Euler angles) also the used convention should be stated. Nevertheless most used notations (matrices and quaternions) are unambiguous.

Figure: Earth Centred Earth Fixed and East, North, Up coordinates.

Tait–Br y an an g le s ar e of te n u sed to de sc r ibe a v eh ic le’s attitu de w i t h respect to a chosen reference frame, though any other notation can b e us ed . Th e p o s i t i v e x-axis in vehicles points always in the direction of movement. For positive y- and z-axis, we have to face two different conventions: • In case of land vehicles like cars, tanks etc., which use th e ENU-system (East-NorthUp) as external r efer enc e (world frame), the vehicle’s positive y- or pitch axis always points to its left, and the positive z- or yaw axis always points up. • By contrast, in case of air and sea vehicles like submarines, ships, airplanes etc., which use the NED-system (North-East-Down) as external reference (world f ra me ), the vehicle’s positive y- or pitch axis always points to its right, and its positive z- or yaw axis always points dow n . • Finally, in case of space vehicles like space shuttles etc., a modification of the latter convention is used, where the vehicle’s positive y- or pitch axis again always points to its right, and its positive z- or yaw axis always points down, but “down” now may have two different meanings: If a so-called local frame is used as external reference, its positive z-axis points “down” to the centre of the earth as it does in case of the earlier mentioned NED-system, but if the inertial frame is used as reference, its positive z-axis will point now to the North Celestial Pole, and its positive x-axis to the Vernal Equinox or some other reference meridian.

Frames Mounted on Vehicles Specially for aircraft, these frames do not need to agree with the earth-bound frames in the up-down line. It must be agreed what ENU and NED mean in this context. Conventions for Land Vehicles

Figure: RPY angles of cars and other land vehicles

For land vehicles is rare to describe their complete orientation, except when speaking

about electronic stability control or satellite navigation. In this case, the convention is normally the one of the adjacent drawing, where RPY stands for roll-pitch-yaw. Conventions for Sea Vehicles

Figure: RPY angles of ships and other sea vehicles

As well as aircraft, the same terminology is used for the motion of ships and boats. It is interesting to note that some words commonly used were introduced in maritime navigation. For example, the yaw angle or heading, has a nautical origin, with the meaning of “bending out of the course”. Etymologically, it is related with the verb ‘to go’. It is related to the concept of bearing. It is typically assigned the shorthand notation Ψ. Conventions for Aircraft Local Reference Frames Coordinates to describe an aircraft attitude (Heading, Elevation and Bank) are normally given relative to a reference control frame located in a control tower, and therefore ENU, relative to the position of the control tower on the earth surface. Coordinates to describe observations made from an aircraft are normally given relative to its intrinsic axes, but normally using as positive the coordinate pointing downwards, where the interesting points are located. Therefore they are normally NED. These axes are normally taken so that × axis is the longitudinal axis pointing ahead, Z axis is the vertical axis pointing downwards, and the Y axis is the lateral one, pointing in such a way that the frame is right handed. The motion of an aircraft is often described in terms of rotation about these axes, so rotation about the X-axis is called rolling, rotation about the Y-axis is called pitching, and rotation about the Z-axis is called yawing. Frames for Space Navigation For satellites orbiting the earth it is normal to use the Equatorial coordinate system. The projection of the Earth’s equator onto the celestial sphere is called the celestial equator. Similarly, the projections of the Earth’s north and south geographic poles become the north and south celestial poles, respectively. Deep space satellites use other Celestial coordinate system, like the Ecliptic coordinate system. Conventions for Space Ships as Aircraft If the goal is to keep the shuttle during its orbits in a constant attitude with respect to the sky, e.g. in order to perform certain astronomical observations, the preferred reference is the inertial frame, and the RPY angle vector (0|0|0) describes an attitude then, where the shuttle’s wings are kept permanently parallel to the earth’s equator, its nose points permanently to the

vernal equinox, and its belly towards the Northern polar star. (Note that rockets and missiles more commonly follow the conventions for aircraft where the RPY angle vector (0|0|0) points north, rather than towards the vernal equinox). On the other hand, if it’s the goal to keep the shuttle during its orbits in an constant attitude with respect to the surface of the earth, the preferred reference will be the local frame, with the RPY angle vector (0|0|0) describing an attitude, where the shuttle’s wings are parallel to the earth’s surface, its nose points to its heading, and its belly down towards the centre of the earth. Frames Used to Describe Attitudes Normally the frames used to describe a vehicle’s local observations are the same frames used to describe its attitude respect the ground tracking stations. An important case in which this does not apply are aircraft. Aircraft observations are performed downwards and therefore normally NED axes convention applies. Nevertheless, when attitudes respect ground stations are given, a relationship between the local earth-bound frame and the onboard ENU frame is used.

Right-hand Rule

Figure: Use of right hand

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century. When choosing three vectors that must be at right angles to each other, there are two distinct solutions, so when expressing this idea in mathematics, one must remove the ambiguity of which solution is meant. There are variations on the mnemonic depending on context, but all variations are related to the one idea of choosing a convention. Direction Associated with an Ordered Pair of Directions One form of the right-hand rule is used in situations in which an ordered operation must be performed on two vectors a and b that has a result which is a vector c perpendicular to both a and b. The most common example is the vector cross product. The right-hand rule imposes the following procedure for choosing one of the two directions. • With the thumb, index, and middle fingers at right angles to each other (with the index

finger pointed straight), the middle finger points in the direction of c when the thumb represents a and the index finger represents b. Other (equivalent) finger assignments are possible. For example, the first (index) finger can represent a, the first vector in the product; the second (middle) finger, b, the second vector; and the thumb, c, the product. Direction Associated with a Rotation

Figure: Prediction of direction of field (B), given that the current I flows in the direction of the thumb

Figure: The right-hand rule as applied to motion produced with screw threads

A different form of the right-hand rule, sometimes called the right-hand grip rule or the corkscrew-rule or the right-hand thumb rule, is used in situations where a vector must be assigned to the rotation of a body, a magnetic field or a fluid. Alternatively, when a rotation is specified by a vector, and it is necessary to understand the way in which the rotation occurs, the right-hand grip rule is applicable. This version of the rule is used in two complementary applications of Ampère’s circuital law: 1. An electric current passes through a solenoid, resulting in a magnetic field. When you wrap your right hand around the solenoid with your fingers in the direction of the conventional current, your thumb points in the direction of the magnetic north pole. 2.

An electric current passes through a straight wire. Here, the thumb points in the direction of the conventional current (from positive to negative), and the fingers point in the direction of the magnetic lines of flux.

The principle is also used to determine the direction of the torque vector. If you grip the imaginary axis of rotation of the rotational force so that your fingers point in the direction of the force, then the extended thumb points in the direction of the torque vector. The right-hand grip rule is a convention derived from the right-hand rule convention for vectors. When applying the rule to current in a straight wire for example, the direction of the magnetic field (counterclockwise instead of clockwise when viewed from the tip of the thumb) is a result of this convention and not an underlying physical phenomenon. Applications

The first form of the rule is used to determine the direction of the cross product of two vectors. This leads to widespread use in physics, wherever the cross product occurs. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.) • The angular velocity of a rotating object and the rotational velocity of any point on the object • A torque, the force that causes it, and the position of the point of application of the force • A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it • A magnetic field in a coil of wire and the electric current in the wire • The force of a magnetic field on a charged particle, the magnetic field itself, and the velocity of the object • The vorticity at any point in the field of flow of a fluid • The induced current from motion in a magnetic field (known as Fleming’s right-hand rule) • The x, y and z unit vectors in a Cartesian coordinate system can be chosen to follow the right-hand rule. Right-handed coordinate systems are often used in rigid body physics and kinematics. Fleming’s left-hand rule is a rule for finding the direction of the thrust on a conductor carrying a current in a magnetic field.

Figure: Fleming’s left-hand rule.

Left-hand Rule In certain situations, it may be useful to use the opposite convention, where one of the vectors is reversed and so creates a left-handed triad instead of a right-handed triad. An example of this situation is for left-handed materials. Normally, for an electromagnetic wave, the electric and magnetic fields, and the direction of propagation of the wave obey the righthand rule. However, left-handed materials have special properties, notably the negative refractive index. It makes the direction of propagation point in the opposite direction. De Graaf’s translation of Fleming’s left-hand rule - which uses thrust, field and current and the right-hand rule, is the FBI rule. The FBI rule changes thrust into F (Lorentz force), B (direction of the magnetic field) and I (current). The FBI rule is easily remembered by US citizens because of the commonly known abbreviation for the Federal Bureau of Investigation. Symmetry Vector

Right-

Right-

Right-

Left-

Left-

Left-

hand

hand

hand

hand

hand

Hand

a, × or I

Thumb

Fingers or palm

First or Index

Thumb

Fingers or palm

First or index

b, y or B

First or index

Thumb

Fingers or palm

Fingers or palm

First or index

Thumb

c, z or F

Fingers or palm

First or index

Thumb

First or index

Thumb

Fingers or palm

In Two Dimensions Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane. The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the “first” and the y-axis the “second” axis) is considered the positive or standard orientation, also called the right-handed orientation. A commonly used mnemonic for defining the positive orientation is the right hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system. The other way of orienting the axes is following the left hand rule, placing the left hand on the plane with the thumb pointing up. When pointing the thumb away from the origin along an axis, the curvature of the fingers indicates a positive rotation along that axis. Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation. In Three Dimensions Once the x- and y-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called ‘right-handed’ and ‘left-handed’. The standard orientation, where the xy-plane is horizontal and the z-axis points up (and the x- and the y-axis form a positively oriented two-dimensional coordinate system in the xy-plane if observed from above the xy-plane) is called right-handed or positive. The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis. Conversely, if the same is done with the left hand, a left-handed system results. Figure depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point towards the observer, whereas the

“middle” axis is meant to point away from the observer. The red circle is parallel to the horizontal xy-plane and indicates rotation from the x-axis to the y-axis (in both cases). Hence the red arrow passes in front of the z-axis. Figure is another attempt at depicting a righthanded coordinate system. Again, there is an ambiguity caused by projecting the threedimensional coordinate system into the plane. This corresponds to the two possible orientations of the coordinate system. Thus the “correct” way to view is to imagine the x-axis as pointing towards the observer and thus seeing a concave corner. Representing a Vector in the Standard Basis A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as r. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as: r = xi + yj where i = (1, 0), and j = (0, 1)are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z)can be written as: r = xi + yj + zk where k = (0, 0, 1) is the unit vector in the direction of the z-axis. There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the complex number whose square is the real number –1 and is identified with the point with coordinates (0,1), so it is not the unit vector in the direction of the x-axis (this confusion is just an unfortunate historical accident). Since the complex numbers can be multiplied giving another complex number, this identification provides a means to “multiply” vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.

Chapter 7: Affine Coordinate System In mathematics, an affine coordinate system is a coordinate system on an affine space where each coordinate is an affine map to the number line. In other words, it is an injective affine map from an affine space A to the coordinate space kn, where k is the base field, for example, the real numbers R. The most important case of affine coordinates in Euclidean spaces is real-valued Cartesian coordinate system. Non-orthogonal affine coordinates are referred to as oblique. A system of n coordinates on n-dimensional space is defined by a (n+ 1)-tuple (O, R1, … Rn ) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula: Note that Rj - O are difference vectors with the origin in O and ends in Rj. An affine space cannot have a coordinate system with n less than its dimension, but n may indeed be greater, which means that the coordinate map is not necessary surjective. Examples of n-coordinate system in an (n-1)-dimensional space are barycentric coordinates and affine homogeneous coordinates (1, x1, … xn-1). In the latter case the x coordinat is equal to 1 on all space, but this “reserved” coordinate facilitates matrix representation of affine maps.

Alpha-numeric Grid An alphanumeric grid (also known as atlas grid) is a simple coordinate system on a grid in which each cell is identified by a combination of a letter and a number. An advantage over numeric coordinates, which use two numbers instead of a number and a letter to refer to a grid cell, is that there can be no confusion over which coordinate refers to which direction. Algebraic chess notation uses an alphanumeric grid to refer to the squares of a chessboard. a

b

c

d

e

f

1

a1

b1

c1

d1

e1

f1

2

a2

b2

c2

d2

e2

f2

3

a3

b3

c3

d3

e3

f3

4

a4

b4

c4

d4

e4

f4

5

a5

b5

c5

d5

e5

f5

6

a6

b6

c6

d6

e6

f6

Astronomical Coordinate Systems Astronomical coordinate systems are coordinate systems used in astronomy to describe the location of objects in the sky and in the universe. The most commonly occurring such systems are coordinate systems on the celestial sphere, but extragalactic coordinates systems are also important for describing more distant objects. Coordinate Systems on the Celestial Sphere

• • • •

Horizontal coordinate system Equatorial coordinate system - based on Earth rotation Ecliptic coordinate system - based on Solar System rotation Galactic coordinate system - based on Milky Way rotation

Bipolar Coordinates Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The other system is two-centre bipolar coordinates. There is also a third coordinate system that is based on two poles (biangular coordinates). The first is based on the Apollonian circles. The curves of constant σ and of τ are circles that intersect at right angles. The coordinates have two foci F1 and F2, which are generally taken to be fixed at (–a, 0) and (a, 0), respectively, on the x-axis of a Cartesian coordinate system. Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates. The bipolar cylindrical coordinates are produced by projecting in the z-direction.

Figure: Bipolar coordinate system

The bispherical coordinates are produced by rotating the bipolar coordinates about the xaxis, i.e., the axis connecting the foci, whereas the toroidal coordinates are produced by rotating the bipolar coordinates about the y-axis, i.e., the axis separating the foci. The classic applications of bipolar coordinates are in solving partial differential equations, e.g., Laplace’s equation or the Helmholtz equation, for which bipolar coordinates allow a separation of variables. A typical example would be the electric field surrounding two parallel cylindrical conductors. The term “bipolar” is sometimes used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such as elliptic coordinates. Definition The most common definition of bipolar coordinates (σ, τ) is

where the σ-coordinate of a point P equals the angle F1 P F2 and the r-coordinate equals the natural logarithm of the ratio of the distances d1 and d2 to the foci

(Recall that F1 and F2 are located at (-a, 0) and (a, 0), respectively.) Equivalently

Curves of Constant a and r Scale Factors The scale factors for the bipolar coordinates (a, r) are equal Thus, the infinitesimal area element equals

and the Laplacian is given by

Other differential operators such as V-Fand VxFcan be expressed in the coordinates (cr, r) by substituting the scale factors into the general formulae found in orthogonal coordinates.

Bipolar Cylindrical Coordinates

Figure: Coordinate surfaces of the bipolar cylindrical coordinates. The yellow crescent corresponds to σ, whereas the red tube corresponds to τ and the blue plane corresponds to z=1. The three surfaces intersect at the point P (shown as a black sphere).

Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular z-direction. The two lines of foci F1 and F2 of the projected Apollonian circles are generally taken to be defined by x = -a and x = +a, respectively, (and by y = 0) in the Cartesian coordinate system. The term “bipolar” is often used to describe other curves

having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g., elliptic coordinates. Basic Definition The most common definition of bipolar cylindrical coordinates (σ,τ, z) is

where the σ coordinate of a point P equals the angle F1PF2 and the τ coordinate equals the natural logarithm of the ratio of the distances d 1 and d2 to the focal lines (Recall that the focal lines F1 and F2 are located at x = -a and x = +a, respectively.) Surfaces of constant σ correspond to cylinders of different radii

that all pass through the focal lines and are not concentric. The surfaces of constant -yare nonintersecting cylinders of different radii

that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the z-axis (the direction of projection). In the z = 0plane, the centres of the constant- σ and constant- cylinders lie on the and axes, respectively. Scale Factors The scale factors for the bipolar coordinates σ and τ are equal

whereas the remaining scale factor hz= 1. Thus, the infinitesimal volume element equals and the Laplacian is given by

Other differential operators such as ∇·Fand ∇×F can be expressed in the coordinates (cr,r)by substituting the scale factors into the general formulae found in orthogonal coordinates. Applications The classic applications of bipolar coordinates are in solving partial differential equations, e.g., Laplace’s equation or the Helmholtz equation, for which bipolar coordinates allow a separation of variables. A typical example would be the electric field surrounding two parallel cylindrical conductors.

Bispherical Coordinates

Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis joining its two foci. The foci are located at distance 1 from the vertical z-axis. The red self-interecting torus is the a=45° isosurface, the blue sphere is the x=0.5 isosurface, and the yellow half-plane is the cp=60° isosurface. The green half-plane marks the x-z plane, from which cp is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239). Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci F and F in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system.

Hyperbolic Coordinates In mathematics, hyperbolic coordinates are a method of locating points in Quadrant I of the Cartesian plane {(x,y):x>0, y >0 } = Q Hyperbolic coordinates take values in the hyperbolic plane defined as: HP = {(u, v) :u ∈ ℝ,v > 0} These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For (x, y) in Q take

and Sometimes the parametre u is called hyperbolic angle and v the geometric mean. The inverse mapping is

This is a continuous mapping, but not an analytic function. Quadrant Model of Hyperbolic Geometry The correspondence affords the hyperbolic geometry structure to Q that is erected on HP by hyperbolic motions. The hyperbolic lines in Q are rays from the origin or petal-shaped curves leaving and reentering the origin. The left-right shift in HP corresponds to a squeeze mapping applied to Q. Note that hyperbolas in Q do not represent geodesics in this model. If one only considers the Euclidean topology of the plane and the topology inherited by Q, then the lines bounding Q seem close to Q. Insight from the metric space HP shows that the open set Q has only the origin as boundary when viewed as the quadrant model of the hyperbolic plane. Indeed, consider rays from the origin in Q, and their images, vertical rays from the boundary R of HP. Any point in HP is an infinite distance from the point p at the foot of the perpendicular to R, but a sequence of points on this perpendicular may tend in the direction of p. The corresponding sequence in Q tends along a ray towards the origin. The old Euclidean boundary of Q is irrelevant to the quadrant model. Applications in Physical Science Physical unit relations like: • V = I R : Ohm’s law • P = V I : Electrical power • P V = k T : Ideal gas law • f λ = c : Sine waves All suggest looking carefully at the quadrant. For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, an isobaric process may trace a hyperbola in the quadrant of absolute temperature and gas density. Statistical Applications •

Comparative study of population density in the quadrant begins with selecting a reference nation, region, or urban area whose population and area are taken as the point (1,1). • Analysis of the elected representation of regions in a representative democracy begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant. Economic Applications There are many natural applications of hyperbolic coordinates in economics: Analysis of currency exchange rate fluctuation: The unit currency sets × = 1. The price currency corresponds to y. For 00 0° ≤ θ ≤ 180° (π rad)

0° ≤ (φ 360° (2π rad) However, the azimuth φ is often restricted to the interval (-180°, +180°], or (-π, +π] in radians, instead of [0, 360°). This is the standard convention for geographic longitude. The range [0°, 180°] for inclination is equivalent to [-90°, +90°] for elevation (latitude). Even with these restrictions, if θ is zero or 180° (elevation is 90° or -90°) then the azimuth angle is arbitrary; and if r is zero, both azimuth and inclination/ elevation are arbitrary. To make the coordinates unique one can use the convention that in these cases the arbitrary coordinates are zero. Plotting To plot a point from its spherical coordinates (r, θ, ϕ), where θ is inclination, move r units from the origin in the zenith direction, rotate by θ about the origin towards the azimuth reference direction, and rotate by ϕ about the zenith in the proper direction. Applications The geographic coordinate system uses the azimuth and elevation of the spherical coordinate system to express locations on Earth, calling them respectively longitude and latitude. Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as rotational matrices. Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as volume integrals inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet’s atmosphere. A sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates. Two important partial differential equations that arise in many physical problems, Laplace’s equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics. Another application is ergonomic design, where r is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. Three dimensional modelling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend towards omnidirectionality at lower frequencies. The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player’s position.

Coordinate System Conversions As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate

system and others. Cartesian Coordinates The spherical coordinates (radius r, inclination θ, azimuth φ) of a point can be obtained from its Cartesian coordinates (x, y, z) by the formulae

The inverse tangent denoted in φ = arctan(y/x) must be suitably defined, taking into account the correct quadrant of (x,y). Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian x-y plane from (x,y) to (R,φ), where R is the projection of r onto the x-y plane, and the second in the Cartesian z-R plane from (z,R) to (r,θ). The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions. These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian x-y plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ=+90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cosθa nd sin θ below become switched. Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth (φ), where r ∈ [0, ∞ ),φ ∈ [0, 2π ],θ ∈ [0, π ], by:

Geographic Coordinates To a first approximation, the geographic coordinate system uses elevation angle (latitude) in degrees north of the equator plane, in the range -90° ≤ φ ≤ 90°, instead of inclination. Latitude is either geocentric latitude, measured at the Earth’s centre and designated variously by ψ, q, φ’, φ, φ or geodetic latitude, measured by the observer’s local vertical, and commonly designated φ. The azimuth angle (longitude), commonly denoted by X, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is -180° < X < 180°. For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation. The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography. Instead of the radial distance, geographers commonly use altitude above some reference surface, which may be the sea level or “mean” surface level for planets without liquid oceans. The radial distance r can be computed from the altitude by adding the mean radius of

the planet’s reference surface, which is approximately 6,360±11 km for Earth. However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometres. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km) and many other details. In astronomy there are a series of spherical coordinate systems that measure the elevation angle from different fundamental planes. These reference planes are the observer’s horizon, the celestial equator (defined by the Earths rotation), the plane of the ecliptic (defined by Earth’s orbit around the sun), and the galactic equator (defined by the rotation of the galaxy). Kinematics In spherical coordinates the position of a point is written, its velocity is then, and its acceleration is,

In the case of a constant ϕ this reduces to vector calculus in polar coordinates.

Homogeneous Coordinates

Figure: Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red)

In mathematics, homogeneous coordinates, introduced by August Ferdinand Mcpbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a

matrix. If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general, one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point on the projective plane. Introduction The projective plane can be thought of as the Euclidean plane with additional points, so called points at infinity, added. There is a point at infinity for each direction, informally defined as the limit of a point that moves in that direction away from a fixed point. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. A given point (x, y) on the Euclidean plane is identified with two ratios (X/Z, Y/Z), so the point (x, y) corresponds to the triple (X, Y, Z) = (xZ, yZ, Z) where Z ≠ 0. Such a triple is a set of homogeneous coordinates for the point (x, y). Note that, since ratios are used, multiplying the three homogeneous coordinates by a common, non-zero factor does not change the point represented – unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. The equation of a line through the point (a, b) may be written l(x - a) + m(y - b) =0 where l and m are not both 0. In parametric form this can be written x = a + mt, y = b - lt. Let Z=1/t, so the coordinates of a point on the line may be written (a + m/Z, b - l/Z)=((aZ + m)/Z, (bZ l)/Z). In homogeneous coordinates this becomes (aZ + m, bZ - l, Z). In the limit as t approaches infinity, in other words as the point moves away from (a, b), Z becomes 0 and the homogeneous coordinates of the point become (m, -l, 0). So (m, -l, 0) are defined as homogeneous coordinates of the point at infinity corresponding to the direction of the line l(x - a) + m(y - b) = 0. To summarize: Any point in the projective plane is represented by a triple (X, Y, Z), called the homogeneous coordinates of the point, where X, Y and Z are not all 0. • The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor. • Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying by a common factor. • When Z is not 0 the point represented is the point (X/ Z, Y/Z) in the Euclidean plane. • When Z is 0 the point represented is a point at infinity. Note that the triple (0, 0, 0) is omitted and does not represent any point. The origin is represented by (0, 0, 1). •

Notation Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. The use of colons instead of commas, for example (x:y:z)

instead of (x, y, z), emphasizes that the coordinates are to be considered ratios. Brackets, as in [x, y, z] emphasize that multiple sets of coordinates are associated with a single point. Some authors use a combination of colons and brackets, as in [x:y:z]. Homogeneity Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say f(x, y, z), does not determine a function defined on points as with Cartesian coordinates. But a condition f(x, y, z) =0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is a k such that f(Ax,Ay,Az) = A,k f(x,y,z). If a set of coordinates represent the same point as (x, y, z) then it can be written (Xx, Xy, Xz) for some non-zero value of X. Then A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by zk , in other words by defining The resulting function f is a polynomial so it makes sense to extend its domain to triples where z =0. The process can be reversed by setting z =1, or The equation f(x, y, z) =0 can then be thought of as the homogeneous form of g(x, y) =0 and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line ax + by + c = 0 is ax + by + cz =0. Other Dimensions The discussions in the preceding sections apply analogously to projective spaces other than the plane. So the points on the projective line may be represented by pairs of coordinates (x, y), not both zero. In this case, the point at infinity is (1, 0). Similarly the points in projective n-space are represented by (n + 1)-tuples. Alternative Definition Another definition of projective space can be given in terms of equivalence classes. For non-zero element of R3, define (x1, y1, z 1 ) ~ (x2, y2, z2) to mean there is a non-zero X so that (x1, y1, z^ = (Xx2, Xy2, Xz2). Then ~ is an equivalence relation and the projective plane can be defined as the equivalence classes of R3 “ {0}. If (x, y, z) is one of elements of the equivalence class p then these are taken to be homogeneous coordinates of p. Lines in this space are defined to be sets of solutions of equations of the form ax + by + cz = 0 where not all of a, b and c are zero. The condition ax + by + cz = 0 depends only on the equivalence class of (x, y, z) so the equation defines a set of points in the projective line. The mapping (x, y) → (x, y, 1) defines an inclusion from the Euclidean plane to the projective plane and the complement of the image is the set of points with z=0. This is the equation of a

line according to the definition and the complement is called the line at infinity. The equivalence classes, p, are the lines through the origin with the origin removed. The origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in R3 that pass through the origin and the coordinates of a non-zero element (x, y, z) of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane. Again, this discussion applies analogously to other dimensions. So the projective space of dimension n can be defined as the set of lines through the origin in Rn+1. Elements Other Than Points The equation of a line in the projective plane may be given as sx + ty + uz = 0 where s, t and u are constants. Each triple (s, t, u) determines a line, the line determined is unchanged if it is multiplied by a nonzero scalar, and at least one of s, t and u must be non-zero. So the triple (s, t, u) may be taken to be homogeneous coordinates of a line in the projective plane, that is line coordinates as opposed to point coordinates. If in sx + ty + uz = 0 the letters s, t and u are taken as variables and x, y and z are taken as constants then equation becomes an equation of a set of lines in the space of all lines in the plane. Geometrically it represents the set of lines that pass though the point (x, y, z) and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may be given sets of four homogeneous coordinates, and so on for higher dimensions.

Duality (Projective Geometry) A striking feature of projective planes is the “symmetry” of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language (the Principle of Duality) and the other a more functional approach. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries which is called a duality. In specific examples, such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite dimensional projective geometry. Principle of Duality If one defines a projective plane axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines, then one may define a plane dual structure. Interchange the role of “points” and “lines” in C=(P,L,I)

to obtain the dual structure C* =(L,P,I*), where I* is the inverse relation of I. C* is also a projective plane, called the dual plane of C. If C and C* are isomorphic, then C is called self-dual. The projective planes PG(2,K) for any division ring K are self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes. In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words “point” and “line” and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of “Two points are on a unique line.” is “Two lines meet at a unique point.” Forming the plane dual of a statement is known as dualizing the statement. If a statement is true in a projective plane C, then the plane dual of that statement must be true in the dual plane C*. This follows since dualizing each statement in the proof “in Cgives a statement of the proof “in C*.” The Principle of Plane Duality says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C. The above concepts can be generalized to talk about space duality, where the terms “points” and “planes” are interchanged (and lines remain lines). This leads to the Principle of Space Duality. Further generalization is possible. These principles provide a good reason for preferring to use a “symmetric” term for the incidence relation. Thus instead of saying “a point lies on a line” one should say “a point is incident with a line” since dualizing the latter only involves interchanging point and line (“a line is incident with a point”). Traditionally in projective geometry, the set of points on a line are considered to include the relation of projective harmonic conjugates. In this tradition the points on a line form a projective range, a concept dual to a pencil of lines on a point. Dual Theorems As the real projective plane, PG(2,R), is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are: • Desargues’ theorem ⇔ Converse of Desargues’ theorem • Pascal’s theorem ⇔ Brianchon’s theorem • Menelaus’ theorem ⇔ Ceva’s theorem

Duality as a Mapping A (plane) duality is a map from a projective plane C = (P,L,I) to its dual plane C* = (L,P,I*) which preserves incidence. That is, a (plane) duality a will map points to lines and lines to points (Pa = L and La = P) in such a way that if a point Q is on a line m ( denoted by Q I m) then Qn I* mn Ô! mn I Qn. A (plane) duality which is an isomorphism is called a

correlation.[1] The existence of a correlation means that the projective plane C is self-dual. In the special case that the projective plane is of the PG(2,K) type, with K a division ring, a duality is called a reciprocity. These planes are always self-dual. By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity, then the reciprocity is called a projective correlation. A correlation of order two (an involution) is called a polarity. If a correlation cp is not a polarity then cp2 is a nontrivial collineation. This duality mapping concept can also be extended to higher dimensional spaces so the modifier “(plane)” can be dropped in those situations. Higher Dimensional Duality Duality in the projective plane is a special case of duality for projective spaces, transformations of PG(n,K) (also denoted by KPn) with K a field, that interchange objects of dimension r with objects of dimension n - 1 - r ( = codimension r + 1). That is, in a projective space of dimension n, the points (dimension 0) are made to correspond with hyperplanes (codimension 1), the lines joining two points (dimension 1) are made to correspond with the intersection of two hyperplanes (codimension 2), and so on. The points of PG(n,K) can be taken to be the nonzero vectors in the (n + 1)-dimensional vector space over K, where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of n-dimensional projective space are the lines through the origin in Kn + 1, which are 1-dimensional vector subspaces. Also the n- vector dimensional subspaces of Kn + 1 represent the (n - 1)- geometric dimensional hyperplanes of projective nspace over K. A nonzero vector u = (u0,u1, ..., u ) in Kn dimensional subspace (hyperplane) Hu, by

+ 1

also determines an (n - 1) - geometric

When a vector u is used to define a hyperplane in this way it shall be denoted by uH, while if it is designating a point we will use uP . In terms of the usual dot product, Hu = {xP : uH • xP = 0}. Since K is a field, the dot product is symmetrical, meaning uH•xP = u0x0 + u1x1 + ... + unxn = x0u0 + x1u1 + ... + xnun = xH•uP . A reciprocity can be given by uP ↔ Hu between points and hyperplanes. This extends to a reciprocity between the line generated by two points and the intersection of two such hyperplanes, and so forth. In the projective plane, PG(2,K), with K a field we have the reciprocity given by: points in homogeneous coordinates (a,b,c) ↔ lines with equations ax + by + cz = 0. In a corresponding projective space, PG(3,K), a reciprocity is given by: points in homogeneous coordinates (a,b,c,d) ↔ planes with equations ax + by + cz + dw = 0. This reciprocity would also map a line determined by two points (a1, b1, c1, d1) and (a2, b2, c2, d2) to the line which is the intersection of the two planes with equations a1x + b1y + c1z + d1w = 0 and a2x + b2y + c2z + d2w = 0. Three Dimensions

In a polarity of real projective 3-space, PG(3,R), points correspond to planes, and lines correspond to lines. By restriction the duality of polyhedra in solid geometry is obtained, where points are dual to faces, and sides are dual to sides, so that the icosahedron is dual to the dodecahedron, and the cube is dual to the octahedron. Geometric Construction of a Reciprocity The reciprocity of PG(2,K), with K a field, given by homogeneous coordinates can also be described geometrically. This uses the model of the real projective plane which is a “unit sphere with antipodes identifie≤, or equivalently, the model of lines and planes through the origin of the vector space K3. Associate a line through the origin with the unique plane through the origin which is perpendicular (orthogonal) to the line. When, in the model, these lines are considered to be the points and the planes the lines of the projective plane PG(2,K), this association becomes a reciprocity (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centreed at the origin. The lines meet the sphere in antipodal points which must then be identified to obtain a point of the projective plane, and the planes meet the sphere in great circles which are thus the lines of the projective plane. That this association “preserves” incidence is most easily seen from the lines and planes model. A point incident with a line in the projective plane corresponds to a line lying in a plane in the model. Applying the association, the plane becomes a line through the origin perpendicular to the plane it is associated with. This image line is perpendicular to every line of the plane which passes through the origin, in particular the original line (point of the projective plane). All lines that are perpendicular to the original line at the origin lie in the unique plane which is orthogonal to the original line, that is, the image plane under the association. Thus, the image line lies in the image plane and the association preserves incidence. Poles and Polars

Figure: Pole and polar with respect to circle O. P = Q’, q is polar of Q, Q is pole of q.

In the Euclidean plane, fix a circle C with centre O and radius r. For each point P other than O define an image point P so that OP • OP = r2. The mapping defined by P → P is called inversion with respect to circle C. The line through P which is perpendicular to the line OP is called the polar of the point P with respect to circle C. Let m be a line not passing through O. Drop a perpendicular from O to m, meeting m at the point Q (this is the point of m that is closest to O).

The image of Q under inversion with respect to C is called the pole of m. If a point P (different from O) is on a line m (not passing through O) then the pole of m lies on the polar of P and viceversa. The incidence preserving process, in which points and lines are transformed into their polars and poles with respect to C is called reciprocation. In order to turn this process into a reciprocity, the Euclidean plane (which is not a projective plane) needs to be expanded to the extended euclidean plane by adding a line at infinity and points at infinity which lie on this line. In this expanded plane, we define the polar of the point O to be the line at infinity (and O is the pole of the line at infinity), and the poles of the lines through O are the points of infinity where, if a line has slope s (=£ 0) its pole is the infinite point associated to the parallel class of lines with slope -1/s. The pole of the x-axis is the point of infinity of the vertical lines and the pole of the yaxis is the point of infinity of the horizontal lines. The construction of a reciprocity based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane). The reciprocities constructed in this manner are projective correlations of order two, that is, polarities. Mapping the Sphere onto the Plane The unit sphere modulo “1 model of the projective plane is isomorphic (w.r.t. incidence properties) to the planar model: the affine plane extended with a projective line at infinity. To map a point on the sphere to a point on the plane, let the plane be tangent to the sphere at some point which shall be the origin of the plane’s coordinate system (2-D origin). Then construct a line passing through the centre of the sphere (3-D origin) and the point on the sphere. This line intersects the plane at a point which is the projection of the point on the sphere onto the plane (or vice versa). This projection can be used to define a one-to-one onto mapping If points in ℝP2 are expressed in homogeneous coordinates, then

Also, lines in the planar model are projections of great circles of the sphere. This is so because through any line in the plane pass an infinitude of different planes: one of these planes passes through the 3-D origin, but a plane passing through the 3-D origin intersects the sphere along a great circle. As we have seen, any great circle in the unit sphere has a projective point perpendicular to it, which can be defined as its dual. But this point is a pair of antipodal points on the unit sphere, through both of which passes a unique 3-D line, and this line extended past the unit sphere intersects the tangent plane at a point, which means that there is a geometric way to associate a unique point on the plane to every line on the plane, such that the point is the dual of the line.

Plücker Coordinates In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a

way to assign six homogenous coordinates to each line in projective 3-space, P3. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4dimensional space of lines in P3 and points on a quadric in P5 (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe k-dimensional linear subspaces, or flats, in an n-dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control. Geometric Intuition A line L in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points x = (x1,x2,x3) and y = (y1,y2,y3). The vector displacement from x to y is nonzero because the points are distinct, and represents the direction of the line. That is, every displacement between points on L is a scalar multiple of d = y-x. If a physical particle of unit mass were to move from x to y, it would have a moment about the origin. The geometric equivalent is a vector whose direction is perpendicular to the plane containing L and the origin, and whose length equals twice the area of the triangle formed by the displacement and the origin. Treating the points as displacements from the origin, the moment is m = x×y, where “×” denotes the vector cross product. The area of the triangle is proportional to the length of the segment between x and y, considered as the base of the triangle; it is not changed by sliding the base along the line, parallel to itself. By definition the moment vector is perpendicular to every displacement along the line, so d•m = 0, where “•” denotes the vector dot product.

Figure: Displacement and moment of two points on line

Although neither d nor m alone is sufficient to determine L, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between x and y. That is, the coordinates (d:m) = (d1:d2:d3:m1:m2:m3) may be considered homogeneous coordinates for L, in the sense that all pairs (λd:λm), for λ ≠ 0, can be produced by points on L and only L, and any such pair determines a unique line so long as d is not zero and d•m = 0. Furthermore, this approach extends to include points, lines, and a plane “at infinity”, in the sense of projective geometry. Example: Let x = (2,3,7) and y = (2,1,0). Then (d:m) =(0:-2:-7:-7:14:-4). Alternatively, let the equations for points x of two distinct planes containing L be

Then their respective planes are perpendicular to vectors a and b, and the direction of L must be perpendicular to both. Hence we may set d = a×b, which is nonzero because a and b are neither zero nor parallel (the planes being distinct and intersecting). If point x satisfies both plane equations, then it also satisfies the linear combination

That is, m = a b - b a is a vector perpendicular to displacements to points on L from the origin; it is, in fact, a moment consistent with the d previously defined from a and b. Example: Let a0 = 2, a =(- 1,0,0) and b0 = - 7, b =(0,7,-2). Then (d:m) =(0: - 2:- 7:7:14:- 4). Although the usual algebraic definition tends to obscure the relationship, (d:m) are the Plücker coordinates of L. Algebraic Definition In a 3-dimensional projective space, P3, let L be a line containing distinct points × and y with homogeneous coordinates (x0:x1:x2: x 3) and (y0:y1:y2:y3), respectively. Let M be the 4×2 matrix with these coordinates as columns.

Because × and y are distinct points, the columns of M are linearly independent; M has rank 2. Let M’ be a second matrix, with columns x’ and y” a different pair of distinct points on L. Then the columns of M’ are linear combinations of the columns of M; so for some 2×2 nonsingular matrix ∇, In particular, rows i and j of M2 and M are related by

Therefore, the determinant of the left side 2×2 matrix equals the product of the determinants of the right side 2×2 matrices, the latter of which is a fixed scalar, det ∇. Primary Coordinates With this motivation, we define Plücker coordinate pij as the determinant of rows i and j of M,

This implies pii = 0 and pij = -pji, reducing the possibilities to only six (4 choose 2) independent quantities. As we have seen, the sixtuple

is uniquely determined by L, up to a common nonzero scale factor. Furthermore, all six components cannot be zero, because if they were, all 2×2 subdeterminants in M would be zero and the rank of M at most one, contradicting the assumption that × and y are distinct. Thus the Plücker coordinates of L, as suggested by the colons, may be considered homogeneous coordinates of a point in a 5-dimensional projective space. Plücker Map Denote the set of all lines (linear images of P1) in P3 by G13. We thus have a map:

where Dual Coordinates Alternatively, let L be a line contained in distinct planes a and b with homogeneous coefficients (a0: a1:a 2: a3) and (b0:b1:b2:b3), respectively. (The first plane equation is 0 = -k ak xk , for example.) Let N be the 2×4 matrix with these coordinates as rows.

We define dual Plücker coordinate pij as the determinant of columns i and j of N,

Dual coordinates are convenient in some computations, and we can show that they are equivalent to primary coordinates. Specifically, let (ij,k,l) be an even permutation of (0,1,2,3); then Pij = Pkl Geometry To relate back to the geometric intuition, take x0 = 0 as the plane at infinity; thus the coordinates of points not at infinity can be normalized so that x = 1. Then M becomes

and setting x = (x1,x2,x3) and y = (y1,y2,y3), we have d = (p01,p02,p03) and m = (p23,p31,p12). Dually, we have d = (p23,p31,p12) and m = (p01,p02,p03). Point Equations Letting (x0:x1:x2:x3) be the point coordinates, four possible points on a line each have coordinates xi = pij , for j = 0…3. Some of these possible points may be inadmissible because

all coordinates are zero, but since at least one Plücker coordinate is nonzero, at least two distinct points are guaranteed. Bijectivity If (q01:q02:q03:q23:q31:q12) are the homogeneous coordinates of a point in P5, without loss of generality assume that q01 is nonzero. Then the matrix

has rank 2, and so its columns are distinct points defining a line L. When the P5 coordinates, qij , satisfy the quadratic Plücker relation, they are the Plücker coordinates of L. To see this, first normalize q01 to 1. Then we immediately have that for the Plücker coordinates computed from M, pij = qij , except for But if the q satisfy the Plücker relation q23 + q02q31 + q03q12 = 0, then p23 = q23, completing the set of identities. Consequently, a is a surjection onto the algebraic variety consisting of the set of zeros of the quadratic polynomial And since a is also an injection, the lines in P3 are thus in bijective correspondence with the points of this quadric in P5, called the Plücker quadric or Klein quadric. Uses Plücker coordinates allow concise solutions to problems of line geometry in 3dimensional space, especially those involving incidence. Line-line Join In the event that two lines are coplanar but not parallel, their common plane has equation where x = (x1, x2, x3). The slightest perturbation will destroy the existence of a common plane, and nearparallelism of the lines will cause numeric difficulties in finding such a plane even if it does exist. Line-line Meet Dually, two coplanar lines, neither of which contains the origin, have common point To handle lines not meeting this restriction. Plane-line Meet Given a plane with equation

or more concisely 0 = a0x0 + a • x; and given a line not in it with Plücker coordinates (d:m), then their point of intersection is The point coordinates, (x0:x1:x2: x3), can also be expressed in terms of Plücker coordinates as

Point-line Join Dually, given a point (y0:y) and a line not containing it, their common plane has equation The plane coordinates, (a0: a1: a2: a3), can also be expressed in terms of dual Plücker coordinates as

Line Families Because the Klein quadric is in P5, it contains linear subspaces of dimensions one and two (but no higher). These correspond to one- and two-parametre families of lines in P3. For example, suppose L and L’ are distinct lines in P3 determined by points x, y and x’, y’, respectively. Linear combinations of their determining points give linear combinations of their Plücker coordinates, generating a one-parametre family of lines containing L and L’. This corresponds to a one-dimensional linear subspace belonging to the Klein quadric. Lines in Plane If th r ee distin c t an d n o n - par alle l lin e s ar e c o plan ar ; th eir lin ear combinations generate a two-parametre family of lines, all the lines in the plane. This corresponds to a two-dimensional linear subspace belonging to the Klein quadric. Lines Through Point If three distinct and non-coplanar lines intersect in a point, their linear combinations generate a two-parametre family of lines, all the lines through the point. This also corresponds to a two-dimensional linear subspace belonging to the Klein quadric. Ruled Surface A ruled surface is a family of lines that is not necessarily linear. It corresponds to a curve on the Klein quadric. For example, a hyperboloid of one sheet is a quadric surface in P3 ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in P5.

Line Geometry During the nineteenth century, line geometry was studied intensively. In terms of the bijection given above, this is a description of the intrinsic geometry of the Klein quadric. Ray Tracing Line geometry is extensively used in ray tracing application where the geometry and intersections of rays need to be calculated in 3D. An implementation is described in Introduction to Pluecker Coordinates written for the Ray Tracing forum by Thouis Jones. Application to Bézout’s Theorem Bézout’s theorem predicts that the number of points of intersection of two curves is equal to the product of their degrees (assuming an algebraically complete field and with certain conventions followed for counting intersection multiplicities). Bézout’s theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates can be used to locate the point of intersection in this case. Similarly, Bézout’s theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them. For example, y = x2 and x = 0 have only one point of intersection in the finite plane. To find the other point of intersection, convert the equations into homogeneous form, yz = x2 and x = 0. This produces x = yz =0 and, assuming not all of x, y and z are 0, the solutions are x = y = 0, z ≠ 0 and x = z =0, y ≠ 0. This first solution is the point (0, 0) in Cartesian coordinates, the finite point of intersection. The second solutions gives the homogeneous coordinates (0, 1, 0) which corresponds to the direction of the y-axis. For the equations xy =1 and x = 0 there are no finite points of intersection. Converting the equations into homogeneous form gives xy = z2 and x = 0. Solving produces the equation z2 = 0 which has a double root at z =0. From the original equation, x = 0, so y ≠ 0 since at least one coordinate must be non-zero. Therefore (0, 1, 0) is the point of intersection counted with multiplicity 2 in agreement with the theorem. Circular points at infinity In projective geometry, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are (1: i: 0) and (1: - i: 0). Here the coordinates are homogeneous coordinates (x: y: z); so that the line at infinity is defined by z =0. These points at infinity are called circular points at infinity because they lie on the complexification of every real circle. In other words, both points satisfy the homogeneous equations of the type The case where the coefficients are all real gives the equation of a general circle (of the real projective plane). In general, an algebraic curve that passes through these two points is called circular. The circular points at infinity are the points at infinity of the isotropic lines.

The circular points are invariant under translation and rotation. Change of Coordinate Systems Just as the selection of axes in the Cartesian coordinate is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore it is useful to know how the different systems are related to each other. Let (x, y, z) be the homogeneous coordinates of a point in the projective plane and for a fixed matrix

with det(A) * 0, define a new set of coordinates (X, Y, Z) by the equation

Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for points in the projective plane. If z is fixed at 1 then are proportional to the signed distances from the point to the lines (The signed distance is the distance multiplied a sign 1 or -1 depending on which side of the line the point lies.) Note that for a = b =0 the value of X is simply a constant, and similarly for Y and Z. The three lines, ax + by + cz = 0, dx + ey + fz = 0, gx + hy + iz = 0 in homogeneous coordinates, or X = 0,Y = 0,Z = 0 in the (X, Y, Z) system, form a triangle called the triangle of reference for the system.

Manifold In mathematics (specifically in geometry and topology), a manifold is a geometrical object that, near each point, resembles Euclidean space of a fixed dimension, called the dimension of the manifold. In mathematical terms, a manifold of dimension n is a topological space such that each point has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles (but not figure eights) are one-dimensional manifolds (1manifolds). The plane, the sphere and the torus are examples of 2-manifolds. Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it

can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

Motivational Examples Circle After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top half of the unit circle, x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this semicircle can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper semicircle to the open interval (– 1,1):

Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle.

Figure: The four charts each map part of the circle to an open interval, and together cover the whole circle.

The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts %top and %ri ht each map this part into the interval (0, 1). Thus a function T from (0, 1) to itself can be constructed, which

first uses the inverse of the top chart to reach the circle and then follows the right chart back to the interval. Let a be any number in (0, 1), then:

Such a function is called a transition map.

Figure: A circle manifold chart based on slope, covering all but one point of the circle. y

The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts and Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (-1, 0); t is the mirror image, with pivot point (+1, 0). The inverse mapping from s to (x, y) is given by

It can easily be confirmed that x2 + y2 = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with

Each chart omits a single point, either (-1, 0) for s or (+1, 0) for t, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and “gluing” the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. Other Curves Manifolds need not be connected (all in “one piece”); an example is a pair of separate circles. In this example we see that a manifold need not have any well-defined notion of distance, for there is no way to define the distance between points that don’t lie in the same

piece. Manifolds need not be closed; thus a line segment without its end points is a manifold. And they are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola (two open, infinite pieces) and the locus of points on a cubic curve y2 = x3-x (a closed loop piece and an open, infinite piece). However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a “+”, not a line (a + is not homeo-morphic to a closed interval (line segment) since deleting the centre point from the + gives a space with four components (i.e. pieces) whereas deleting a point from a closed interval gives a space with at most two pieces; topological operations always preserve the number of pieces). Enriched Circle Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable. The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transition functions have these properties as well. Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold. History The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Early Development Before the modern concept of a manifold there were several important results. NonEuclidean geometry considers spaces where Euclid’s parallel postulate fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai, and Riemann developed them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces,

V -E+F = 2. The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topological map with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope. Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its ‘parallel’ and ‘meridian’ circles, creating a map with V = 1 vertex, E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 - 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss– Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Synthesis Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables. Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Simeon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology. Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann’s original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as “manifoldness”. In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannig-faltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colours and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n-1) dimensional manifoldnesses. Riemann’s intuitive notion of a Mannig-faltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann.

Topology of Manifolds: Highlights Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture. Bill Thurston’s geometrization programme, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang-Mills theory), where they serve as a substitute for ordinary ‘flat’ space-time. Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higherdimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory. Mathematical definition Informally, a manifold is a space that is “modeled on” Euclidean space. There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. In terms of constructing manifolds via patching, a manifold has an additional structure if the transition maps between different patches satisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighbourhoods diffeomorphic with each other, so that the manifold has a welldefined set of functions which are differentiable in each neighbourhood, and so differentiable on the manifold as a whole. Formally, a topological manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space. Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense ‘too large’ such as the long line, while Hausdorff excludes spaces such as “the line with two origins” (these generalizations of manifolds are discussed in non-Hausdorff manifolds). Locally homeomorphic to Euclidean space means that every point has a neighbourhood homeomorphic to an open Euclidean n-ball, Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions. If a manifold has a fixed dimension, it is called a pure manifold. For example, the sphere has a constant

dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected component has a fixed dimension. Schemetheoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. Broad Definition The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. This omits the point-set axioms, allowing higher cardinalities and non-Hausdorff manifolds; and it omits finite dimension, allowing structures such as Hilbert manifolds to be modeled on Hilbert spaces, Banach manifolds to be modeled on Banach spaces, and Fréchet manifolds to be modeled on Fréchet spaces. Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

Charts, Atlases, and Transition Maps The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map’s boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. Charts A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R2 minus the positive x-axis and the origin. Another example of a chart is the map χtop mentioned in the section above, a chart for the circle. Atlases The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple

ways using different combinations of charts. Two atlases are said to be Ck equivalent if their union is also a Ck atlas. The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas. Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is an abstract object and not used directly (e.g. in calculations). Transition Maps Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in Rn to the manifold and then back to another (or perhaps the same) open ball in Rn. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map. Additional Structure An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all the transition maps are compatible with this structure, the structure transfers to the manifold. This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of Rn (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms. The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible. These notions are made precise in general through the use of pseudogroups. Construction A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Charts

Figure: The chart maps the part of the sphere with positive z coordinate to a disc.

Perhaps the simplest way to construct a manifold is the one used in the example above of

the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere: Sphere with Charts A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R3: The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R2. Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere. This can be easily generalized to higher-dimensional spheres. Patchwork A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold. This can be illustrated with the transition map t = 1D from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point s = 1D t on the first copy (the points t = 0 and s = 0 are not identified with any point on the first and second copy, respectively). This gives a circle. Intrinsic and Extrinsic View The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to

some surface through that point. The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. n-Sphere as a Patchwork The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2sphere) to higher dimensions. An n-sphere Sn can be constructed by gluing together two copies of Rn. The transition map between them is defined as This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier. Identifying Points of a Manifold It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold is the real projective space identified as a quotient space of the corresponding sphere. One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifold and G is the group, the resulting quotient space is denoted by M / G (or G \ M). Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively). Gluing Along Boundaries Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together. Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For other manifolds other structures should be preserved. A finite cylinder may be constructed as a manifold by starting with a strip [0, 1] × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries.

Manifolds with additional structure Topological Manifolds The simplest kind of manifold to define is the topological manifold, which looks locally like some “ordinary” Euclidean space Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rn. These homeomorphisms are the charts of the manifold. It is to be noted that a topological manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular and consistent choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles. Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable. The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number n in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds. Differentiable Manifolds For most applications a special kind of topological manifold, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series). The sphere can be given analytic structure, as can most familiar curves and surfaces. There are also topological manifolds, i.e., locally Euclidean spaces, which possess no differentiable structures at all. A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

Riemannian Manifolds To measure distances and angles on manifolds, the manifold must be Riemannian. A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product 0Å-, Å- 0 in a manner which varies smoothly from point to point. Given two tangent vectors u and v, the inner product0u,v 0gives a real number. The dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields. All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it. Finsler Manifolds A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product. Any Riemannian manifold is a Finsler manifold. Lie Groups Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps. A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as U(1), can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation. Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n by n matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n2-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are n(n-1)/2 dimensional manifolds, where n-1 is the dimension of the sphere. Further examples can be found in the table of Lie groups. Other Types of Manifolds •

A complex manifold is a manifold modeled on Cn with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. Note

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that an n-dimensional complex manifold has dimension 2n as a real differentiable manifold. A CR manifold is a manifold modeled on boundaries of domains in Cn. Infinite dimensional manifolds: to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces. A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold. A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. A digital manifold is a special kind of combinatorial manifold which is defined in digital space.

Classification and Invariants Different notions of manifolds have different notions of classi-fication and invariant; in this section we focus on smooth closed manifolds. The classification of smooth closed manifolds is well-understood in principle, except in dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the Solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants. This is much harder in higher dimensions: higher dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object. However, one can determine if two manifolds are different if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are invariant under different descriptions. Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no programme exists that can decide whether two manifolds are diffeomorphic. Smooth manifolds have a rich set of invariants, coming from point-set topology, classic

algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between no local invariants and local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global. Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the simply connected property and orientability. Indeed several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds. Genus and the Euler Characteristic For two dimensional manifolds a key invariant property is the genus, or the “number of handles” present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed it is possible to fully characterize compact, twodimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology and cohomology. Maps of Manifolds Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuous functions and smooth functions generally, there are maps with special properties. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem. Generalizations of Manifolds • Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of “singularities” in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense. • Manifold with corners • Algebraic varieties and schemes: Non-singular algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, an algebraic variety is glued together from affine algebraic

varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using sheaves instead of atlases. Because of singular points, a variety is in general not a manifold, though linguistically the French variété, German Mannigfaltigkeit and Eng lish ma ni fo ld ar e la rg el y synonymous. In French an algebraic variety is called une variété algébrique (an algebraic variety), while a smooth manifold is called une variété différentielle (a differential variety). •

CW-complexes: A CW complex is a topological spaceformed by gluing disks of different dimensionalitytogether. In general the resulting space is singular, and hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory, as they are easy to compute with and singularities are not a concern.

Change of Coordinates In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as “local”) coordinate system, fixed to the node, is defined based on the first (typically referred to as “global” or “world” coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix, which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three unit vectors aligned with those axes. Transformations A coordinate transformation is a conversion from one system to another, to describe the same space. With every bijection from the space to itself two coordinate transformations can be associated: •

Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation) • Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation) For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to -3, so that the coordinate of each point becomes 3 more. Systems Commonly Used Some coordinate systems are the following:

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The Cartesian coordinate system (also called the “rectangular coordinate system”), which, for two- and three-dimensional spaces, uses two and three numbers (respectively) representing distances from the origin in three mutually perpendicular directions. • Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves. • Polar coordinate system represents a point in the plane by a distance from the origin and an angle measured from a reference line intersecting the origin. • Log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin. • Cylindrical coordinate system represents a point in three-space using two perpendicular axes; distance is measured along one axis, while the other axis formes the reference line for a polar coordinate representation of the remaining two components. • Spherical coordinate system represents a point in three space by the distance from the origin and two angles measured from two reference lines which intersect the origin. • Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates. • Generalized coordinates are used in the Lagrangian treatment of mechanics. • Canonical coordinates are used in the Hamiltonian treatment of mechanics. • Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines. • Barycentric coordinates (mathematics) as used for Ternary_plot There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include: • Whewell equation relates arc length and tangential angle. • Cesàro equation relates arc length and curvature.

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