# Multivariable Mathematics: Linear Algebra, Multivariable, Calculus, and Manifolds

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English Pages [505] Year 2004

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
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
Glossary
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Index