VECTOR CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL FORMS A Unified Approach STH EDITION JOHN H. HUBBARD BARBARA BURKE HUBBARD sgn( a-) signature of a permutation (Theorem and Definition 4.8.11) Span span (Definition 2.4.3) (J" standard deviation (Definition 3.8.6). Also denotes permutation. sum (Section 0.1) sup supremum; least upper bound (Definitions 0.5.1, 1.6.5) Supp(!) support of a function f (Definition 4.1.2) T (tau) torsion (Definition 3.9.14) TxX tangent space to manifold (Definition 3.2.1) tr trace of a matrix (Definition 1.4.13) v·w dot product of two vectors, (Definition 1.4.1) vxw cross product of two vectors (Definition 1.4.17) lvl length of vector v (Definition 1.4.2) (v)J_ orthogonal complement to subspace spanned by v (proof of Theorem 3.7.15) Varf variance (Definitions 3.8.6) [x]k k-truncation (Definition Al.2) Notation particular to this book [OJ matrix with all entries 0 (equation 1. 7.48) equal in the sense of Lebesgue (Definition 4.11.6) L ~_(e.g., vi) "hat" indicating omitted factor (equation 6.5.25) A result of row reducing A to echelon form (Theorem 2.1.7) [Alb] matrix formed from columns of A and b (sentence after equation 2.1. 7) Br(x) ball of radius r around x (Definition 1.5.1) /Jn volume of unit ball (Example 4.5. 7) DN(Ilr) dyadic paving (Definition 4.1. 7) [Df(a)] derivative off at a (Proposition and Definition 1.7.9) IIJ) set of finite decimals (Definition Al.4) D1f higher partial derivatives (equation 3.3.11) Wxl integrand for multiple integral (Section 4.1); see also Section 5.3 8'AfX smooth part of the boundary of X (Definition 6.6.2) r(J) graph off (Definition 3.1.1) [h]R R-truncation of h (equation 4.11.26) <Pf flux form (Definition 6.5.2) <j) {v} concrete to abstract function (Definition 2.6.12) I~ set of multi-exponents (Notation 3.3.5) [Jf(a)] Jacobian matrix (Definition 1. 7. 7) L(J) lower integral (Definition 4.1.10) mA(f) infimum of f(x) for x EA (Definition 4.1.3) Mt mass form (Definition 6.5.4) MA(!) supremum of f(x) for x EA (Definition 4.1.3) Mat(n,m) space of n x m matrices (Discussion before Proposition 1.5.38) OSCA(f) oscillation off over A (Definition 4.1.4) 0 orientation (Definition 6.3.1) olv} orientation specified by { v} (paragraph after Definition 6.3.1) ost standard orientation (Section 6.3) Pf,a Taylor polynomial (Definition 3.3.13) P(v1, ... vk) k-parallelogram (Definition 4.9.3) Px(v1, ... vk) anchored k-parallelogram (Section 5.1) [Pv1~v] change of basis matrix (Proposition and Definition 2.6.17) Qn, Q unit n-dimensional cube (Definition 4.9.4) UOK (equation 5.2.18) U(f) upper integral (Definition 4.1.10) v and v column vector and point (Definition 1.1.2) v0ln n-dimensional volume (Definition 4.1.17) Wf work form (Definition 6.5.1) x,y used in denoting tangent space (paragraph before Example 3.2.2) NOTATION For Greek letters, see Section 0.1; for set theory notation, Section 0.3. Numbers give first use. Those preceded by "A" are in appendix A. Standard and semi-standard notation def equal by definition ---> to (discussion following Definition 0.4.1) maps to (margin note near Definition 0.4.1) lA indicator function (Definition 4.1.1) (a, b) open interval, also denoted ]a, b[ [a,b] closed interval IAI length of matrix A (Definition 1.4.10) llAll norm of a matrix A (Definition 2.9.6) jT transpose (paragraph before Definition 0.4.5) A-1 inverse (Proposition and Definition 1.2.14) A~(llr) space of constant k-forms in nr (Definition 6.1.7) Ak(U) space of k-form fields on U (Definition 6.1.16) :B magnetic field (Example 6.5.13) c cone operator from Ak(U) to Ak-1(U) (Definition 6.13.9) c cone over parallelogram (Definition 6.13. 7) ci once continuously differentiable (Definition 1.9.6) GP p times continuously differentiable (Definition 1.9. 7) c2 the space of C2 functions (Example 2.6.7) c closure of C (Definition 1.5.8) c0 interior of C (Definition 1.5.9) C(O, 1) space of continuous real-valued functions on (0, 1) (Example 2.6.2) corr correlation (Definitions 3.8.6) COY covariance (Definition 3.8.6) d exterior derivative from Ak(U) to Ak+1(U) (Definition 6.7.1) 8A boundary of A (Definition 1.5.10) detA determinant of A (Definition 1.4.13) dim dimension (Proposition and Definition 2.4.21) Dif partial derivative; also denoted jL (Definition 1.7.3) de~~ted Dj(Dif) second partial derivative, also a 828! . (Definition 2 .8. 7) Xj Xi ei, ... ,en standard basis vectors (Definition 1.1. 7) E elementary matrix (Definition 2.3.5) :E electric field (Example 6.5.13) J og composition (Definition 0.4.12) f (k) kth derivative off (line before Theorem 3.3.1) j Fourier transform off (Definition 4.11.24) r,r positive, negative part of function (Definition 4.1.15) lF Faraday 2-form (Example 6.5.13) H mean curvature (Definition 3.9.7) I identity matrix (Definition 1.2.10) img image (Definition 0.4.2) inf infimum; greatest lower bound (Definition 1.6.7) K (kappa) curvature of a curve (Definitions 3.9.1, 3.9.14) K Gaussian curvature (Definition 3.9.8) ker kernel (Definition 2.5.1) M Maxwell 2-form (equation 6.12.8) lnx natural logarithm of x (i.e., loge x) '\7 nabla, also called "de!" (Definition 6.8.1) 0 little o (Definition 3.4.1) 0 big o (Definition All.1) IT product (Section 0.1) S1, S" unit circle, unit sphere (Example 1.1.6, Example 5.3.14) VECTOR CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL FORMS A UNIFIED APPROACH 5TH EDITION John Hamal Hubbard Barbara Burke Hubbard CORNELL UNIVERSITY UNIVERSITE AIX-MARSEILLE MATRIX EDITIONS ITHACA, NY 14850 MATRIXEDITIONS.COM Matrix Editions The Library of Congress has cataloged the 4th edition as follows: Hubbard, John H. Vector calculus, linear algebra, and differential forms : a unified approach / John Hamal Hubbard, Barbara Burke Hubbard. - 4th ed. p. cm. Includes bibliographical references and index. ISBN 978-0-9715766-5-0 (alk. paper) 1. Calculus. 2. Algebras, Linear. I. Hubbard, Barbara Burke, 1948- II. Title. QA303.2.H83 2009 515' .63-dc22 2009016333 Copyright 2015 by Matrix Editions 214 University Ave. Ithaca, NY 14850 Matrix Editions www.MatrixEditions.com All rights reserved. This book may not be translated, copied, or reproduced, in whole or in part, in any form or by any means, without written permission from the publisher, except for brief excerpts in connection with reviews or scholarly analysis. Printed in the United States of America 10987654321 ISBN 978-0-9715766-8-1 Cover image: The Wave by the American painter Albert Bierstadt (1830-1902). A breaking wave is just the kind of thing to which a physicist would want to apply Stokes's theorem: the balance between surface tension (a surface integral) and gravity and momentum (a volume integral) is the key to the cohesion of the water. The crashing of the wave is poorly understood; we speculate that it corresponds to the loss of the conditions to be a piece-with-boundary (see Section 6.6). The crest carries no surface tension; when it acquires positive area it breaks the balance. In this picture, this occurs rather suddenly as you travel from left to right along the wave, but a careful look at real waves will show that the picture is remarkably accurate. A Adrien et Regine Douady, pour l 'inspimtion d 'une vie entiere Contents PREFACE vii CHAPTER 0 PRELIMINARIES 0.0 Introduction 1 0.1 Reading mathematics 1 0.2 Quantifiers and negation 4 0.3 Set theory 6 0.4 Functions 9 0.5 Real numbers 17 0.6 Infinite sets 22 0. 7 Complex numbers 25 CHAPTER 1 VECTORS, MATRICES, AND DERIVATIVES LO Introduction 32 1.1 Introducing the actors: Points and vectors 33 1.2 Introducing the actors: Matrices 42 1.3 Matrix multiplication as a linear transformation 56 1.4 The geometry of Rn 67 1.5 Limits and continuity 83 1.6 Five big theorems 104 1. 7 Derivatives in several variables as linear transformations 119 1.8 Rules for computing derivatives 137 1.9 The mean value theorem and criteria for differentiability 145 1.10 Review exercises for Chapter 1 152 CHAPTER 2 SOLVING EQUATIONS 2.0 Introduction 159 2.1 The main algorithm: Row reduction 160 2.2 Solving equations with row reduction 166 2.3 Matrix inverses and elementary matrices 175 2.4 Linear combinations, span, and linear independence 180 2.5 Kernels, images, and the dimension formula 192 2.6 Abstract vector spaces 207 2.7 Eigenvectors and eigenvalues 219 2.8 Newton's method 232 2.9 Superconvergence 252 2.10 The inverse and implicit function theorems 258 2.11 Review exercises for Chapter 2 277 iv Contents v CHAPTER 3 MANIFOLDS, TAYLOR POLYNOMIALS, QUADRATIC FORMS, AND CURVATURE 3.0 Introduction 283 3.1 Manifolds 284 3.2 Tangent spaces 305 3.3 Taylor polynomials in several variables 314 3.4 Rules for computing Taylor polynomials 325 3.5 Quadratic forms 332 3.6 Classifying critical points of functions 342 3.7 Constrained critical points and Lagrange multipliers 349 3.8 Probability and the singular value decomposition 367 3.9 Geometry of curves and surfaces 378 3.10 Review exercises for Chapter 3 396 CHAPTER 4 INTEGRATION 4.0 Introduction 401 4.1 Defining the integral 402 4.2 Probability and centers of gravity 417 4.3 What functions can be integrated? 424 4.4 Measure zero 430 4.5 Fubini's theorem and iterated integrals 438 4.6 Numerical methods of integration 449 4.7 Other pavings 459 4.8 Determinants 461 4.9 Volumes and determinants 479 4.10 The change of variables formula 486 4.11 Lebesgue integrals 498 4.12 Review exercises for Chapter 4 520 CHAPTER.5 VOLUMES OF MANIFOLDS 5.0 Introduction 524 5.1 Parallelograms and their volumes 525 5.2 Parametrizations 528 5.3 Computing volumes of manifolds 538 5.4 Integration and curvature 550 5.5 Fractals and fractional dimension 560 5.6 Review exercises for Chapter 5 562 CHAPTER 6 FORMS AND VECTOR CALCULUS 6.0 Introduction 564 6.1 Forms on !Rn 565 6.2 Integrating form fields over parametrized domains 577 6.3 Orientation of manifolds 582 vi Contents 6.4 Integrating forms over oriented manifolds 589 6.5 Forms in the language of vector calculus 599 6.6 Boundary orientation 611 6. 7 The exterior derivative 626 6.8 Grad, curl, div, and all that 633 6.9 The pullback 640 6.10 The generalized Stokes's theorem 645 6.11 The integral theorems of vector calculus 661 6.12 Electromagnetism 669 6.13 Potentials 688 6.14 Review exercises for Chapter 6 699 APPENDIX: ANALYSIS A.O Introduction 704 A.1 Arithmetic of real numbers 704 A.2 Cubic and quartic equations 708 A.3 Two results in topology: Nested compact sets and Heine-Borel 713 A.4 Proof of the chain rule 715 A.5 Proof of Kantorovich's theorem 717 A.6 Proof of Lemma 2.9.5 (superconvergence) 723 A. 7 Proof of differentiability of the inverse function 724 A.8 Proof of the implicit function theorem 729 A.9 Proving the equality of crossed partials 732 A.10 Functions with many vanishing partial derivatives 733 A.11 Proving rules for Taylor polynomials; big 0 and little o 735 A.12 Taylor's theorem with remainder 740 A.13 Proving Theorem 3.5.3 (completing squares) 745 A.14 Classifying constrained critical points 746 A.15 Geometry of curves and surfaces: Proofs 750 A.16 Stirling's formula and proof of the central limit theorem 756 A.17 Proving Fubini's theorem 760 A.18 Justifying the use of other pavings 762 A.19 Change of variables formula: A rigorous proof 765 A.20 Volume 0 and related results 772 A.21 Lebesgue measure and proofs for Lebesgue integrals 776 A.22 Computing the exterior derivative 794 A.23 Proving Stokes's theorem 797 BIBLIOGRAPHY 804 PHOTO CREDITS 805 INDEX 807 Preface . . . The numerical interpretation . . . is however necessary. . . . So long as it is not obtained, the solutions may be said to remain in complete and useless, and the truth which it is proposed to discover is no less hidden in the formulae of analysis than it was in the physical problem itself. -Joseph Fourier, The Analytic Theory of Heat Joseph Fourier (1768-1830) Fourier was arrested during the French Revolution and threatened Chapters 1 through 6 of this book cover the standard topics in multivariate with the guillotine, but survived calculus and a first course in linear algebra. The book can also be used for and later accompanied Napoleon a course in analysis, using the proofs in the Appendix. to Egypt; in his day he was as well The organization and selection of material differs from the standard ap- known for his studies of Egypt as proach in three ways, reflecting the following principles. for his contributions to mathemat ics and physics. He found a way First, we believe that at this level linear algebra should be more a con to solve linear partial differential venient setting and language for multivariate calculus than a subject equations while studying heat dif in its own right. The guiding principle of this unified approach is that locally, a nonlinear function behaves like its derivative. fusion. An emphasis on computa tionally effective algorithms is one When we have a question about a nonlinear function we answer it by theme of this book. looking carefully at a linear transformation: its derivative. In this approach, everything learned about linear algebra pays off twice: first for understand ing linear equations, then as a tool for understanding nonlinear equations. We discuss abstract vector spaces in Section 2.6, but the emphasis is on !Rn, as we believe that most students find it easiest to move from the concrete to the abstract. Second, we emphasize computationally effective algorithms, and we prove theorems by showing that these algorithms work. We feel this better reflects the way mathematics is used today, in applied and pure mathematics. Moreover, it can be done with no loss of rigor. For linear equations, row reduction is the central tool; we use it to prove all the standard results about dimension and rank. For nonlinear equations, the cornerstone is Newton's method, the best and most widely used method for solving nonlinear equations; we use it both as a computational tool and in proving the inverse and implicit function theorems. We include a section on numerical methods of integration, and we encourage the use of computers both to reduce tedious calculations and as an aid in visualizing curves and surfaces. Third, we use differential forms to generalize the fundamental theorem of calculus to higher dimensions. vii