Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 5th edition

complex Analysis Is One Of The Most Attractive Of All The Core Topics In An Undergraduate Mathematics Course. Its Importance To Applications Means That It Can Be Studied Both From A Very Pure Perspective And A Very Applied Perspective. This Book Takes Account Of These Varying Needs And Backgrounds And Provides A Self-study Text For Students In Mathematics, Science And Engineering. Beginning With A Summary Of What The Student Needs To Know At The Outset, It Covers All The Topics Likely To Feature In A First Course In The Subject, Including: Complex Numbers, Differentiation, Integration, Cauchy's Theorem, And Its Consequences, Laurent Series And The Residue Theorem, Applications Of Contour Integration, Conformal Mappings, And Harmonic Functions. A Brief Final Chapter Explains The Riemann Hypothesis, The Most Celebrated Of All The Unsolved Problems In Mathematics, And Ends With A Short Descriptive Account Of Iteration, Julia Sets And The Mandelbrot Set. Clear And Careful Explanations Are Backed Up With Worked Examples And More Than 100 Exercises, For Which Full Solutions Are Provided. Cover Contents Preface Acknowledgments Preliminaries 0.0 Introduction 0.1 Reading mathematics 0.2 Quantifiers and negation Exercise for Section 0.2 0.3 Set theory Exercise for Section 0.3 0.4 Functions Exercises for Section 0.4 0.5 Real numbers Exercises for Section 0.5 0.6 Infinite sets Exercises for Section 0.6 0.7 Complex numbers Exercises for Section 0.7 Vectors, matrices, and derivatives 1.0 Introduction 1.1 Introducing the actors: Points and vectors Exercises for Section 1.1 1.2 Introducing the actors: Matrices Exercises for Section 1.2 1.3 What the actors do: Matrix multiplication as a linear transformation Exercises for Section 1.3 1.4 The geometry of Exercises for Section 1.4 1.5 Limits and continuity Exercises for Section 1.5 1.6 Five big theorems Exercises for Section 1.6 1.7 Derivatives in several variables as linear transformations Exercises for Section 1.7 1.8 Rules for computing derivatives Exercises for Section 1.8 1.9 The mean value theorem and criteria for differentiability Exercises for Section 1.9 1.10 Review exercises for Chapter 1 Solving equations 2.0 Introduction 2.1 The main algorithm: row reduction Exercises for Section 2.1 2.2 Solving equations with row reduction Exercises for Section 2.2 2.3 Matrix inverses and elementary matrices Exercises for Section 2.3 2.4 Linear combinations, span, and linear independence Exercises for Section 2.4 2.5 Kernels, images, and the dimension formula Exercises for Section 2.5 2.6 Abstract vector spaces Exercises for Section 2.6 2.7 Eigenvectors and eigenvalues Exercises for Section 2.7 2.8 Newton’s method Exercises for Section 2.8 2.9 Superconvergence Exercises for Section 2.9 2.10 The inverse and implicit function theorems Exercises for Section 2.10 2.11 Review exercises for Chapter 2 Manifolds, Taylor polynomials, quadratic forms, and curvature 3.0 Introduction 3.1 Manifolds Exercises for Section 3.1 3.2 Tangent spaces Exercises for Section 3.2 3.3 Taylor polynomials in several variables Exercises for Section 3.3 3.4 Rules for computing Taylor polynomials Exercises for Section 3.4 3.5 Quadratic forms Exercises for Section 3.5 3.6 Classifying critical points of functions Exercises for Section 3.6 3.7 Constrained critical points and Lagrange multipliers Exercises for Section 3.7 3.8 Probability and the singular value decomposition Exercises for Section 3.8 3.9 Geometry of curves and surfaces Exercises for Section 3.9 3.10 Review exercises for Chapter 3 Integration 4.0 Introduction 4.1 Defining the integral Exercises for Section 4.1 4.2 Probability and centers of gravity Exercises for Section 4.2 4.3 What functions can be integrated? Exercises for Section 4.3 4.4 Measure zero Exercises for Section 4.4 4.5 Fubini’s theorem and iterated integrals Exercises for Section 4.5 4.6 Numerical methods of integration Exercises for Section 4.6 4.7 Other pavings Exercises for Section 4.7 4.8 Determinants Exercises for Section 4.8 4.9 Volumes and determinants Exercises for Section 4.9 4.10 The change of variables formula Exercises for Section 4.10 4.11 Lebesgue integrals Exercises for Section 4.11 4.12 Review exercises for Chapter 4 Volumes of manifolds 5.0 Introduction 5.1 Parallelograms and their volumes Exercises for Section 5.1 5.2 Parametrizations Exercises for Section 5.2 5.3 Computing volumes of manifolds Exercises for Section 5.3 5.4 Integration and curvature Exercises for Section 5.4 5.5 Fractals and fractional dimension Exercises for Section 5.5 5.6 Review exercises for Chapter 5 Forms and vector calculus 6.0 Introduction 6.1 Forms on Exercises for Section 6.1 6.2 Integrating form fields over parametrized domains Exercises for Section 6.2 6.3 Orientation of manifolds Exercises for Section 6.3 6.4 Integrating forms over oriented manifolds Exercises for Section 6.4 6.5 Forms in the language of vector calculus Exercises for Section 6.5 6.6 Boundary orientation Exercises for Section 6.6 6.7 The exterior derivative Exercises for Section 6.7 6.8 Grad, curl, div, and all that Exercises for Section 6.8 6.9 The pullback Exercises for Section 6.9 6.10 The generalized Stokes’s theorem Exercises for Section 6.10 6.11 The integral theorems of vector calculus Exercises for Section 6.11 6.12 Electromagnetism Exercises for Section 6.12 6.13 Potentials Exercises for Section 6.13 6.14 Review exercises for Chapter 6 A0 Introduction Appendix: Analysis A1 Arithmetic of real numbers A2 Cubic and quartic equations A3 Two results in topology: Nested compact sets and Heine-Borel A4 Proof of the chain rule A5 Proof of Kantorovich’s theorem A6 Proof of Lemma 2.9.5 (superconvergence) A7 Proof of differentiability of the inverse function A8 Proof of the implicit function theorem A9 Proving the equality of crossed partials A10 Functions with many vanishing partial derivatives A11 Proving rules for Taylor polynomials A12 Taylor’s theorem with remainder A13 Proving Theorem 3.5.3 (completing squares) A14 Classifying constrained critical points A15 Geometry of curves and surfaces: Proofs A16 Stirling’s formula and the central limit theorem A17 Proving Fubini’s theorem A18 Justifying the use of other pavings A19 The change of variables formula: A rigorous proof A20 Volume 0 and related results A21 Lebesgue measure and proofs for Lebesgue integrals A22 Computing the exterior derivative A23 Proof of Proposition 6.10.10 (for Stokes’s theorem) Bibliography Index This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. Complex analysis can be a difficult subject and many introductory texts are just too ambitious for today’s students. This book takes a lower starting point than is traditional and concentrates on explaining the key ideas through worked examples and informal explanations, rather than through "dry" theory.