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

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
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Index