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Vector Calculus, Linear Algebra, and Differential Forms:
A Unified Approach
Table of Contents
Preface
Chapter 0 Preliminaries
0.0 Introduction
0.1 Reading Mathematics
0.2 Quantifiers and Negation
0.3 Set Theory
0.4 Functions
0.5 Real Numbers
0.6 Infinite Sets
0.7 Complex Numbers
Chapter 1 Vectors, Matrices, and Derivatives
1.0 Introduction
1.1 Introducing the Actors: Points and Vectors
1.2 Introducing the Actors: Matrices
1.3 A Matrix as a Transformation
1.4 The Geometry of Rn
1.5 Limits and Continuity
1.6 Four Big Theorems
1.7 Differential Calculus
1.8 Rules for Computing Derivatives
1.9 Mean Value Theorem and Criteria for Differentiability
1.10 Review Exercises
Chapter 2 Solving Equations
2.0 Introduction
2.1 The Main Algorithm: Row Reduction
2.2 Solving Equations Using Row Reduction
2.3 Matrix Inverses and Elementary Matrices
2.4 Linear Combinations, Span, and Linear Independence
2.5 Kernels, Images, and the Dimension Formula
2.6 An Introduction to Abstract Vector Spaces
2.7 Newton's Method
2.8 Superconvergence
2.9 The Inverse and Implicit Function Theorems
2.10 Review Exercises
Chapter 3 Higher Partial Derivatives, Quadratic Forms, and Manifolds
3.0 Introduction
3.1 Manifolds
3.2 Tangent Spaces
3.3 Taylor Polynomials in Several Variables
3.4 Rules for Computing Taylor Polynomials
3.5 Quadratic Forms
3.6 Classifying Critical Points of Functions
3.7 Constrained Critical Points and Lagrange Multipliers
3.8 Geometry of Curves and Spaces
3.9 Review Exercises
Chapter 4 Integration
4.0 Introduction
4.1 Defining the Integral
4.2 Probability and Centers of Gravity
4.3 What Functions Can Be Integrated?
4.4 Integration and Measure Zero
4.5 Fubini's Theorem and Iterated Integrals
4.6 Numerical Methods of Integration
4.7 Other Pavings
4.8 Determinants
4.9 Volumes and Determinants
4.10 The Change of Variables Formula
4.11 Lebesgue Integrals
4.12 Review Exercises
Chapter 5 Volumes of Manifolds
5.0 Introduction
5.1 Parallelograms and their Volumes
5.2 Parametrizations
5.3 Computing Volumes of Manifolds
5.4 Fractals and Fractional Dimension
5.5 Review Exercises
Chapter 6 Forms and Vector Calculus
6.0 Introduction
6.1 Forms on Rn
6.2 Integrating Form Fields over Parametrized Domains
6.3 Orientation of Manifolds
6.4 Integrating Forms over Oriented Manifolds
6.5 Forms and Vector Calculus
6.6 Boundary Orientation
6.7 The Exterior Derivative
6.8 The Exterior Derivative in the Language of Vector Calculus
6.9 The Generalized Stokes's Theorem
6.10 The Integral Theorems of Vector Calculus
6.11 Potentials
6.12 Review Exercises
Appendix
A.0 Introduction
A.1 Arithmetic of Real Numbers
A.2 Cubic and Quartic Equations
A.3 Two Extra Results in Topology
A.4 Proof of the Chain Rule
A.5 Proof of Kantorovich's Theorem
A.6 Proof of Lemma 2.8.5 (Superconvergence)
A.7 Differentiability of the Inverse Function
A.8 Proof of the Implicit Function Theorem
A.9 Proof of Theorem 3.3.9: Equality of Crossed Partials
A.10 Proof of Proposition 3.3.19
A.11 Proof of Rules for Taylor Polynomials
A.12 Taylor's Theorem with Remainder
A.13 Proof of Theorem 3.5.3 (Completing Squares)
A.14 Geometry of Curves and Surfaces: Proofs
A.15 Proof of the Central Limit Theorem
A.16 Proof of Fubini's Theorem
A.17 Justifying the Use of Other Pavings
A.18 Existence and Uniqueness of the Determinant
A.19 Rigorous Proof of the Change of Variables Formula
A.20 Justifying Volume 0
A.21 Lebesgue Measure and Proofs for Lebesgue Integrals
A.22 Justifying the Change of Parametrization
A.23 Computing the Exterior Derivative
A.24 The Pullback
A.25 Proof of Stokes' Theorem
