Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

Description

Contents CHAPTER 1 VECTORS, MATRICES, DERIVATIVES 1.1 Introducing the actors: Points and vectors 1.2 Introducing the actors: Matrices 1.3 What the actors do: Matrix multiplication as a linear transformation 1.4 The geometry of Rn 1.5 Limits and continuity 1.6 Five big theorems 1.7 Derivatives in several variables as linear transformations 1.8 Rules for computing derivatives 1.9 The mean value theorem and criteria for differentiability 1.10 Review exercises for Chapter 1 CHAPTER 2 SOLVING EQUATIONS 2.1 The main algorithm: Row reduction 2.2 Solving equations with row reductions 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 Abstract vector spaces 2.7 Eigenvectors and eigenvalues 2.8 Newton's method 2.9 Superconvengence 2.10 The inverse and implicit function theorems 2.11 Review exercises for chapter 2 CHAPTER 3 MANIFOLDS, TAYLOR POLYNOMIALS, QUADRATIC FORMS, AND CURVATURE 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 Probability and the singular value decomposition 3.9 Geometry of curves and surfaces 3.10 Review exercises for Chapter 3 CHAPTER 4 INTEGRATION 4.1 Defining the integral 4.2 Probability and centers of gravity 4.3 What functions can be integrated 4.4 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 for Chapter 4 CHAPTER.5 VOLUMES OF MANIFOLDS 5.1 Parallelograms and their volumes 5.2 Parametrizations 5.3 Computing volumes of manifolds 5.4 Integration and curvature 5.5 Fractals and fractional dimension 5.6 Review exercises for Chapter 5 CHAPTER 6 FORMS AND VECTOR CALCULUS 6.1 Forms on !Rn 6.2 Integrating form fields over parametrized domains 6.3 Orientation of manifolds vi Contents 6.4 Integrating forms over oriented manifolds 6.5 Forms in the language of vector calculus 6.6 Boundary orientation 6. 7 The exterior derivative 6.8 Grad, curl, div, and all that 6.9 The pullback 6.10 The generalized Stokes's theorem 6.11 The integral theorems of vector calculus 6.12 Electromagnetism 6.13 Potentials 6.14 Review exercises for Chapter 6 APPENDIX: ANALYSIS A.O Introduction A.1 Arithmetic of real numbers A.2 Cubic and quartic equations A.3 Two results in topology: Nested compact sets and Heine-Borel A.4 Proof of the chain rule A.5 Proof of Kantorovich's theorem A.6 Proof of Lemma 2.9.5 (superconvergence A. 7 Proof of differentiability of the inverse function A.8 Proof of the implicit function theorem A.9 Proving the equality of crossed partials A.10 Functions with many vanishing partial derivatives A.11 Proving rules for Taylor polynomials; big 0 and little o A.12 Taylor's theorem with remainder A.13 Proving Theorem 3.5.3 (completing squares A.14 Classifying constrained critical points A.15 Geometry of curves and surfaces: Proofs A.16 Stirling's formula and proof of the central limit theorem A.17 Proving Fubini's theorem A.18 Justifying the use of other pavings A.19 Change of variables formula: A rigorous proof A.20 Volume 0 and related results A.21 Lebesgue measure and proofs for Lebesgue integrals A.22 Computing the exterior derivative A.23 Proving Stokes's theorem BIBLIOGRAPHY PHOTO CREDITS INDEX