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All the Mathematics You MissedBeginning graduate students in mathematics and other quantitativesubjects are expected to have a daunting breadth of mathematicalknowledge, but few have such a background. This book will helpstudents see the broad outline of mathematics and to fill in the gaps intheir knowledge.The author explains the basic points and a few key results of the mostimportant undergraduate topics in mathematics, emphasizing theintuitions behind the subject. The topics include linear algebra, vectorcalculus, differential geometry, real analysis, point-set topology,differential equations, probability theory, complex analysis, abstractalgebra, and more. An annotated bibliography offers a guide to furtherreading and more rigorous foundations.This book will be an essential resource for advanced undergraduateand beginning graduate students in mathematics, the physical sciences,engineering, computer science, statistics, and economics, and for anyoneelse who needs to quickly learn some serious mathematics.Thomas A. Garrity is Professor of Mathematics at Williams College inWilliamstown, Massachusetts. He was an undergraduate at theUniversity of Texas, Austin, and a graduate student at Brown University,receiving his Ph.D. in 1986. From 1986 to 1989, he was G.c. EvansInstructor at Rice University. In 1989, he moved to Williams College,where he has been ever since except in 1992-3, when he spent the year atthe University of Washington, and 2000-1, when he spent the year at theUniversity of Michigan, Ann Arbor.All the Mathematics You MissedBut Need to Know for Graduate SchoolThomas A. GarrityWilliams CollegeFigures by Lori PedersenCAMBRIDGEUNIVERSITY PRESSPUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITYOFCAMBRIIX:;EThe Pitt Building, Trumpington Street, Cambridge, United KingdomCAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA10 Stamford Road, Oakleigh, VIC 3166, AustraliaRuiz de Alarcon 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africahttp://www.cambridge.org Thomas A Garrity 2002This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published 2002Printed in the United States of AmericaTypeface Palatino 10/12 pt.A catalog record for this book is available from the British Library.Library of Congress Cataloging in Publication DataGarrity, Thomas A, 1959-All the mathematics you missed: but need to know for graduateschool 1Thomas A Garrity.p. em.Includes bibliographical references and index.ISBN 0-521-79285-1 - ISBN 0-521-79707-1 (pb.)1. Mathematics. 1. TItle.QA37.3 .G37200251D-dc21 2001037644ISBN 0 521 79285 1 hardbackISBN 0 521 79707 1 paperbackDedicated to the MemoryofRobert MiznerContentsPrefaceOn the Structure of MathematicsBrief Summaries of Topics0.1 Linear Algebra .0.2 Real Analysis .0.3 Differentiating Vector-Valued Functions0.4 Point Set Topology . . . . . . . . . . . .0.5 Classical Stokes' Theorems .0.6 Differential Forms and Stokes' Theorem0.7 Curvature for Curves and Surfaces0.8 Geometry . . . . . . . . . . . . . . . .0.9 Complex Analysis ..0.10 Countability and the Axiom of Choice0.11 Algebra .0.12 Lebesgue Integration0.13 Fourier Analysis ..0.14 Differential Equations0.15 Combinatorics and Probability Theory0.16 Algorithms .1 Linear Algebra1.1 Introduction .1.2 The Basic Vector Space Rn .1.3 Vector Spaces and Linear Transformations .1.4 Bases and Dimension .1.5 The Determinant . . . . . . . . . . .1.6 The Key Theorem of Linear Algebra1.7 Similar Matrices .1.8 Eigenvalues and Eigenvectors . . . .xiiixixxxiiiXXlllxxiiixxiiiXXIVXXIVXXIVXXIVXXVXXVXXVIxxvixxviXXVIXXVllXXVllXXVll112469121415Vlll1.9 Dual Vector Spaces .1.10 Books ..1.11 Exercises .....2 E and J Real Analysis2.1 Limits .....2.2 Continuity...2.3 Differentiation2.4 Integration ..2.5 The Fundamental Theorem of Calculus.2.6 Pointwise Convergence of Functions2.7 Uniform Convergence .2.8 The Weierstrass M-Test2.9 Weierstrass' Example.2.10 Books ..2.11 Exercises .3 Calculus for Vector-Valued Functions3.1 Vector-Valued Functions ...3.2 Limits and Continuity . . . . .3.3 Differentiation and Jacobians .3.4 The Inverse Function Theorem3.5 Implicit Function Theorem3.6 Books ..3.7 Exercises ....4 Point Set Topology4.1 Basic Definitions .4.2 The Standard Topology on Rn4.3 Metric Spaces . . . . . . . . . .4.4 Bases for Topologies . . . . . .4.5 Zariski Topology of Commutative Rings4.6 Books ..4.7 Exercises .5 Classical Stokes' Theorems5.1 Preliminaries about Vector Calculus5.1.1 Vector Fields .5.1.2 Manifolds and Boundaries.5.1.3 Path Integrals ..5.1.4 Surface Integrals5.1.5 The Gradient ..5.1.6 The Divergence.CONTENTS202121232325262831353638404344474749505356606063636672737577788182828487919393CONTENTS5.1.7 The Curl .5.1.8 Orientability .5.2 The Divergence Theorem and Stokes' Theorem5.3 Physical Interpretation of Divergence Thm. .5.4 A Physical Interpretation of Stokes' Theorem5.5 Proof of the Divergence Theorem ...5.6 Sketch of a Proof for Stokes' Theorem5.7 Books ..5.8 Exercises .IX9494959798991041081086 Differential Forms and Stokes' Thm. 1116.1 Volumes of Parallelepipeds. . . . . . 1126.2 Diff. Forms and the Exterior Derivative 1156.2.1 Elementary k-forms 1156.2.2 The Vector Space of k-forms .. 1186.2.3 Rules for Manipulating k-forms . 1196.2.4 Differential k-forms and the Exterior Derivative. 1226.3 Differential Forms and Vector Fields 1246.4 Manifolds . . . . . . . . . . . . . . . . . . . . . . . 1266.5 Tangent Spaces and Orientations . . . . . . . . . . 1326.5.1 Tangent Spaces for Implicit and ParametricManifolds . . . . . . . . . . . . . . . . . 1326.5.2 Tangent Spaces for Abstract Manifolds. . . 1336.5.3 Orientation of a Vector Space . . . . . . . . 1356.5.4 Orientation of a Manifold and its Boundary . 1366.6 Integration on Manifolds. 1376.7 Stokes'Theorem 1396.8 Books . . 1426.9 Exercises .... 1437 Curvature for Curves and Surfaces 1457.1 Plane Curves 1457.2 Space Curves . . . . . . . . . 1487.3 Surfaces . . . . . . . . . . . . 1527.4 The Gauss-Bonnet Theorem. 1577.5 Books . . 1587.6 Exercises 1588 Geometry 1618.1 Euclidean Geometry 1628.2 Hyperbolic Geometry 1638.3 Elliptic Geometry. 1668.4 Curvature....... 167x8.5 Books ..8.6 Exercises9 Complex Analysis9.1 Analyticity as a Limit .9.2 Cauchy-Riemann Equations .9.3 Integral Representations of Functions.9.4 Analytic Functions as Power Series9.5 Conformal Maps .9.6 The Riemann Mapping Theorem .9.7 Several Complex Variables: Hartog's Theorem.9.8 Books ..9.9 Exercises .10 Countability and the Axiom of Choice10.1 Countability .10.2 Naive Set Theory and Paradoxes10.3 The Axiom of Choice. . . . . . .lOA Non-measurable Sets .10.5 Godel and Independence Proofs .10.6 Books ..10.7 Exercises .11 Algebra11.1 Groups .11.2 Representation Theory.11.3 Rings .11.4 Fields and Galois Theory11.5 Books ..11.6 Exercises .....12 Lebesgue Integration12.1 Lebesgue Measure12.2 The Cantor Set . .12.3 Lebesgue Integration12.4 Convergence Theorems.12.5 Books ..12.6 Exercises .13 Fourier Analysis13.1 Waves, Periodic Functions and Trigonometry13.2 Fourier Series . . .13.3 Convergence Issues . . . . . . . . . . . . . . .CONTENTS168169171172174179187191194196197198201201205207208210211211213213219221223228229231231234236239241241243243244250.13.4 Fourier Integrals and Transforms13.5 Solving Differential Equations.13.6 Books ..13.7 Exercises .14 Differential Equations14.1 Basics .14.2 Ordinary Differential Equations .14.3 The Laplacian. . . . . . . . . .14.3.1 Mean Value Principle ..14.3.2 Separation of Variables .14.3.3 Applications to Complex Analysis14.4 The Heat Equation .14.5 The Wave Equation .. ..14.5.1 Derivation .14.5.2 Change of Variables14.6 Integrability Conditions14.7 Lewy's Example14.8 Books ..14.9 Exercises ....15 Combinatorics and Probability15.1 Counting .15.2 Basic Probability Theory .15.3 Independence . . . . . . . .15.4 Expected Values and Variance.15.5 Central Limit Theorem ....15.6 Stirling's Approximation for n!15.7 Books ..15.8 Exercises .16 Algorithms16.1 Algorithms and Complexity .16.2 Graphs: Euler and Hamiltonian Circuits16.3 Sorting and Trees. . . . . . . . . . ..16.4 P=NP? .16.5 Numerical Analysis: Newton's Method16.6 Books ..16.7 Exercises .A Equivalence Relations252256258258261261262266266267270270273273277279281282282285285287290291294300305305307308308313316317324324327PrefaceMath is Exciting. We are living in the greatest age of mathematics everseen. In the 1930s, there were some people who feared that the risingabstractions of the early twentieth century would either lead to mathe-maticians work