*371*
*49*
*23MB*

*English*
*Pages [505]*
*Year 2004*

Table of contents :

Contents

Preface

Chapter 1 Vectors and Matrices

1 Vectors in Rn

2 Dot Product

3 Subspaces of Rn

4 Linear Transformations and Matric Algebra

5 Introduction to Determinants and the Cross Product

Chapter 2 Functions, Limits, And Continuity

1 Scalar and Vector Valued Functions

1.1 Parametrized Curves

1.2 Scalar Functions of Several Variables

1.3 Vector Functions of Several Variables

2 A Bit of Topology in Rn

3 Limits and Continuity

Chapter 3 The Derivative

1 Partial Derivatives and Directional Derivatives

2 Differentiability

3 Differentiation Rules

4 The Gradient

5 Curves

6 Higher Order Partial Derivatives

Chapter 4 Implicit and Explicit Solutions of Linear Systems

1 Gaussian Elimination and the Theory of Linear Systems

2 Elementary Matrices and Calculating Inverse Matrices

3 Linear Independence, Basis, And Dimension

3.1 Abstract Vector Spaces

4 The Four Fundamental Subspaces

5 The Nonlinear Case: Introduction to Manifolds

Chapter 5 Extremum Problems

1 Compactness and the Maximum Value Theorem

2 Maximum/Minimum Problems

3 Quadratic Forms and the Second Derivative Test

4 Lagrange Multipliers

5 Projections, Least Squares, and Inner Product Spaces

Chapter 6 Solving Nonlinear Problems

1 The Contraction Mapping Principle

2 The Inverse and Implicit Function Theorems

3 Manifolds Revisited

Chapter 7 Integration

1 Multiple Integrals

2 Iterated Integrals and Fubini's Theorem

3 Polar, Cylindrical, and Spherical Coordinates

4 Physical Applications

5 Determinants and n-Dimensional Volume

6 Change of Variables Theorem

Chapter 8 Differential Forms and Integration on Manifolds

1 Motivation

2 Differential Forms

2.1 The Multilinear Setup

2.2 Differential Forms on Rn and the Exterior Derivative

2.3 Pullback

3 Line Integrals and Green's Theorem

3.1 The Fundamental Theorem of Calculus for Line Integrals

3.2 Finding a Potential Function

3.3 Green's Theorem

4 Surface Integrals and Flux

4.1 Oriented Surfaces in R3 and Flux

4.2 Surface Area

5 Stoke's Theorem

5.1 Integrating over a General Compact, Oriented k-Dimensional Manifold

5.2 Stoke's Theorem

6 Applications to Physics

6.1 The Dictionary in R3

6.2 Gauss's Law

6.3 Maxwell's Equations

7 Applications to Topology

Chapter 9 Eigenvalues, Eigenvectors, and Applications

1 Linear Transformations and Change of Basis

2 Eigenvalues, Eigenvectors, and Diagonalizability

2.1 The Characteristic Polynomial

2.2 Diagonalizability

3 Difference Equations and Ordinary Differential Equations

3.1 Differential Equations

3.2 Systems of Differential Equations

3.3 Flows and the Divergence Theorem

4 The Spectral Theorem

4.1 Conics and Quadric Surfaces

Glossary

Answers

1

2

3

4

5

6

7

8

9

Index

Contents

Preface

Chapter 1 Vectors and Matrices

1 Vectors in Rn

2 Dot Product

3 Subspaces of Rn

4 Linear Transformations and Matric Algebra

5 Introduction to Determinants and the Cross Product

Chapter 2 Functions, Limits, And Continuity

1 Scalar and Vector Valued Functions

1.1 Parametrized Curves

1.2 Scalar Functions of Several Variables

1.3 Vector Functions of Several Variables

2 A Bit of Topology in Rn

3 Limits and Continuity

Chapter 3 The Derivative

1 Partial Derivatives and Directional Derivatives

2 Differentiability

3 Differentiation Rules

4 The Gradient

5 Curves

6 Higher Order Partial Derivatives

Chapter 4 Implicit and Explicit Solutions of Linear Systems

1 Gaussian Elimination and the Theory of Linear Systems

2 Elementary Matrices and Calculating Inverse Matrices

3 Linear Independence, Basis, And Dimension

3.1 Abstract Vector Spaces

4 The Four Fundamental Subspaces

5 The Nonlinear Case: Introduction to Manifolds

Chapter 5 Extremum Problems

1 Compactness and the Maximum Value Theorem

2 Maximum/Minimum Problems

3 Quadratic Forms and the Second Derivative Test

4 Lagrange Multipliers

5 Projections, Least Squares, and Inner Product Spaces

Chapter 6 Solving Nonlinear Problems

1 The Contraction Mapping Principle

2 The Inverse and Implicit Function Theorems

3 Manifolds Revisited

Chapter 7 Integration

1 Multiple Integrals

2 Iterated Integrals and Fubini's Theorem

3 Polar, Cylindrical, and Spherical Coordinates

4 Physical Applications

5 Determinants and n-Dimensional Volume

6 Change of Variables Theorem

Chapter 8 Differential Forms and Integration on Manifolds

1 Motivation

2 Differential Forms

2.1 The Multilinear Setup

2.2 Differential Forms on Rn and the Exterior Derivative

2.3 Pullback

3 Line Integrals and Green's Theorem

3.1 The Fundamental Theorem of Calculus for Line Integrals

3.2 Finding a Potential Function

3.3 Green's Theorem

4 Surface Integrals and Flux

4.1 Oriented Surfaces in R3 and Flux

4.2 Surface Area

5 Stoke's Theorem

5.1 Integrating over a General Compact, Oriented k-Dimensional Manifold

5.2 Stoke's Theorem

6 Applications to Physics

6.1 The Dictionary in R3

6.2 Gauss's Law

6.3 Maxwell's Equations

7 Applications to Topology

Chapter 9 Eigenvalues, Eigenvectors, and Applications

1 Linear Transformations and Change of Basis

2 Eigenvalues, Eigenvectors, and Diagonalizability

2.1 The Characteristic Polynomial

2.2 Diagonalizability

3 Difference Equations and Ordinary Differential Equations

3.1 Differential Equations

3.2 Systems of Differential Equations

3.3 Flows and the Divergence Theorem

4 The Spectral Theorem

4.1 Conics and Quadric Surfaces

Glossary

Answers

1

2

3

4

5

6

7

8

9

Index

- Author / Uploaded
- Theodore Shifrin

Table of Contents Contents ...........................................................................6 Preface ..............................................................................8