All the Mathematics You Missed Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge, but few have such a background. This book will help students see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains thebasic points and a few keyresults ofthe most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential geometry, real analysis, point-set topology, differential equations, probability theory, complex analysis, abstract algebra, and more. An annotated bibliography offers a guide to further readingand more rigorous foundations. This book will be an essential resource for advanced undergraduate and beginning graduate students in mathematics, the physical sciences, engineering, computer science, statistics, and economics, and for anyone else who needs to quicklylearn some serious mathematics. Thomas A. Garrity is Professor of Mathematics at Williams College in Williamstown, Massachusetts. He was an undergraduate at the UniversityofTexas,Austin,and a graduate student at Brown University, receiving his Ph.D. in 1986. From 1986 to 1989, he was G.c. Evans Instructor at Rice University. In 1989, he moved to Williams College, where he has been ever since exceptin 1992-3,when he spent the yearat the University of Washington, and 2000-1, when he spent the year at the UniversityofMichigan,AnnArbor. All the Mathematics You Missed But Need to Know for Graduate School Thomas A. Garrity Williams College FiguresbyLori Pedersen CAMBRIDGE UNIVERSITY PRESS PUBLISHED BYTHE PRESS SYNDICATE OFTHE UNIVERSITY OFCAMBRIIX:;E ThePitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITYPRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West20th Street, New York, NY10011-4211,USA 10Stamford Road, Oakleigh, VIC 3166,Australia Ruiz deAlarcon13, 28014 Madrid, Spain DockHouse, The Waterfront, Cape Town 8001, SouthAfrica http://www.cambridge.org © Thomas A Garrity2002 This book is in copyright. Subjectto statutory exception and to the provisions ofrelevantcollective licensingagreements, no reproduction ofanypartmay take place without the written permission ofCambridgeUniversityPress. First published 2002 Printed in the United States ofAmerica Typeface Palatino 10/12 pt. Acatalogrecordfor this bookis availablefrom the British Library. LibraryofCongress Cataloging in Publication Data Garrity, ThomasA,1959- Allthemathematicsyou missed: butneed to knowfor graduate school1ThomasA Garrity. p. em. Includes bibliographicalreferences and index. ISBN0-521-79285-1 - ISBN0-521-79707-1 (pb.) 1.Mathematics. 1.TItle. QA37.3 .G372002 51D-dc21 2001037644 ISBN0521 792851hardback ISBN0521 797071paperback Dedicated to the Memory of Robert Mizner Contents Preface xiii On the Structure of Mathematics xix BriefSummaries ofTopics xxiii 0.1 Linear Algebra . XXlll 0.2 Real Analysis . xxiii 0.3 Differentiating Vector-Valued Functions xxiii 0.4 Point Set Topology . . . . . . . . . . . . XXIV 0.5 Classical Stokes' Theorems . XXIV 0.6 Differential Forms and Stokes' Theorem XXIV 0.7 Curvature for Curves and Surfaces XXIV 0.8 Geometry . . . . . . . . . . . . . . . . XXV 0.9 Complex Analysis .. XXV 0.10 Countability and the Axiom of Choice XXVI 0.11 Algebra . xxvi 0.12 Lebesgue Integration xxvi 0.13 Fourier Analysis .. XXVI 0.14 Differential Equations XXVll 0.15 Combinatorics and Probability Theory XXVll 0.16 Algorithms . XXVll 1 Linear Algebra 1 1.1 Introduction . 1 1.2 The Basic Vector Space Rn . 2 1.3 Vector Spaces and Linear Transformations . 4 1.4 Bases and Dimension . 6 1.5 The Determinant . . . . . . . . . . . 9 1.6 The Key Theorem ofLinear Algebra 12 1.7 Similar Matrices . 14 1.8 Eigenvalues and Eigenvectors . . . . 15 CONTENTS Vlll 1.9 Dual Vector Spaces . 20 1.10 Books .. 21 1.11 Exercises ..... 21 2 E and J Real Analysis 23 2.1 Limits ..... 23 2.2 Continuity... 25 2.3 Differentiation 26 2.4 Integration .. 28 2.5 The Fundamental Theorem of Calculus. 31 2.6 Pointwise Convergence of Functions 35 2.7 Uniform Convergence . 36 2.8 The Weierstrass M-Test 38 2.9 Weierstrass' Example. 40 2.10 Books .. 43 2.11 Exercises . 44 3 Calculus for Vector-Valued Functions 47 3.1 Vector-Valued Functions ... 47 3.2 Limits and Continuity . . . . . 49 3.3 Differentiation and Jacobians . 50 3.4 The Inverse Function Theorem 53 3.5 Implicit Function Theorem 56 3.6 Books .. 60 3.7 Exercises .... 60 4 Point Set Topology 63 4.1 Basic Definitions . 63 4.2 The Standard Topology on Rn 66 4.3 Metric Spaces . . . . . . . . . . 72 4.4 Bases for Topologies . . . . . . 73 4.5 Zariski Topology of Commutative Rings 75 4.6 Books .. 77 4.7 Exercises . 78 5 Classical Stokes' Theorems 81 5.1 Preliminaries about Vector Calculus 82 5.1.1 Vector Fields . 82 5.1.2 Manifolds and Boundaries. 84 5.1.3 Path Integrals .. 87 5.1.4 Surface Integrals 91 5.1.5 The Gradient .. 93 5.1.6 The Divergence. 93 CONTENTS IX 5.1.7 The Curl . 94 5.1.8 Orientability . 94 5.2 The Divergence Theorem and Stokes' Theorem 95 5.3 Physical Interpretation ofDivergence Thm. . 97 5.4 A Physical Interpretation ofStokes' Theorem 98 5.5 Proofof the Divergence Theorem ... 99 5.6 Sketch ofa Prooffor Stokes' Theorem 104 5.7 Books .. 108 5.8 Exercises . 108 6 Differential Forms and Stokes' Thm. 111 6.1 Volumes ofParallelepipeds. . . . . . 112 6.2 Diff. Forms and the Exterior Derivative 115 6.2.1 Elementary k-forms 115 6.2.2 The Vector Space of k-forms .. 118 6.2.3 Rules for Manipulating k-forms . 119 6.2.4 Differential k-forms and the Exterior Derivative. 122 6.3 Differential Forms and Vector Fields 124 6.4 Manifolds . . . . . . . . . . . . . . . . . . . . . . . 126 6.5 Tangent Spaces and Orientations . . . . . . . . . . 132 6.5.1 Tangent Spaces for Implicit and Parametric Manifolds . . . . . . . . . . . . . . . . . 132 6.5.2 Tangent Spaces for Abstract Manifolds. . . 133 6.5.3 Orientation of a Vector Space . . . . . . . . 135 6.5.4 Orientation of a Manifold and its Boundary . 136 6.6 Integration on Manifolds. 137 6.7 Stokes'Theorem 139 6.8 Books . . 142 6.9 Exercises .... 143 7 Curvature for Curves and Surfaces 145 7.1 Plane Curves 145 7.2 Space Curves . . . . . . . . . 148 7.3 Surfaces . . . . . . . . . . . . 152 7.4 The Gauss-Bonnet Theorem. 157 7.5 Books . . 158 7.6 Exercises 158 8 Geometry 161 8.1 Euclidean Geometry 162 8.2 Hyperbolic Geometry 163 8.3 Elliptic Geometry. 166 8.4 Curvature....... 167