All the mathematics you missed (but need to know for graduate school)

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 the basic points and a few key results of the 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 reading and 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 quickly learn some serious mathematics Thomas A Garrity is Professor of Mathematics at Williams College in Williamstown, Massachusetts He was an undergraduate at the University of Texas, 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 except in 1992-3, when he spent the year at the University of Washington, and 2000-1, when he spent the year at the University of Michigan, Ann Arbor All the Mathematics You Missed But Need to Know for Graduate School Thomas A Garrity Williams College Figures by Lori Pedersen CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIIX:;E The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Thomas A Garrity 2002 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2002 Printed in the United States of America Typeface Palatino 10/12 pt A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication Data Garrity, Thomas A, 1959All the mathematics you missed: but need to know for graduate school Thomas A Garrity p em Includes bibliographical references and index ISBN 0-521-79285-1 - ISBN 0-521-79707-1 (pb.) Mathematics TItle QA37.3 G372002 51D-dc21 2001037644 ISBN 521 79285 hardback ISBN 521 79707 paperback Dedicated to the Memory of Robert Mizner Contents Preface xiii On the Structure of Mathematics xix Brief Summaries of Topics 0.1 Linear Algebra 0.2 Real Analysis 0.3 Differentiating Vector-Valued Functions 0.4 Point Set Topology 0.5 Classical Stokes' Theorems 0.6 Differential Forms and Stokes' Theorem 0.7 Curvature for Curves and Surfaces 0.8 Geometry 0.9 Complex Analysis 0.10 Countability and the Axiom of Choice 0.11 Algebra 0.12 Lebesgue Integration 0.13 Fourier Analysis 0.14 Differential Equations 0.15 Combinatorics and Probability Theory 0.16 Algorithms Linear Algebra 1.1 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 Algebra 1.7 Similar Matrices 1.8 Eigenvalues and Eigenvectors xxiii XXlll xxiii xxiii XXIV XXIV XXIV XXIV XXV XXV XXVI xxvi xxvi XXVI XXVll XXVll XXVll 1 12 14 15 CONTENTS Vlll 1.9 Dual Vector Spaces 1.10 Books 1.11 Exercises 20 21 21 and J Real Analysis 2.1 Limits 2.2 Continuity 2.3 Differentiation 2.4 Integration 2.5 The Fundamental Theorem of Calculus 2.6 Pointwise Convergence of Functions 2.7 Uniform Convergence 2.8 The Weierstrass M-Test 2.9 Weierstrass' Example 2.10 Books 2.11 Exercises 23 23 25 26 28 31 35 36 38 Calculus for Vector-Valued Functions 3.1 Vector-Valued Functions 3.2 Limits and Continuity 3.3 Differentiation and Jacobians 3.4 The Inverse Function Theorem 3.5 Implicit Function Theorem 3.6 Books 3.7 Exercises 47 Point Set Topology 4.1 Basic Definitions 4.2 The Standard Topology on R n 4.3 Metric Spaces 4.4 Bases for Topologies 4.5 Zariski Topology of Commutative Rings 4.6 Books 4.7 Exercises 63 63 66 72 73 Classical Stokes' Theorems 5.1 Preliminaries about Vector Calculus 5.1.1 Vector Fields 5.1.2 Manifolds and Boundaries 5.1.3 Path Integrals 5.1.4 Surface Integrals 5.1.5 The Gradient 5.1.6 The Divergence 81 82 82 E 40 43 44 47 49 50 53 56 60 60 75 77 78 84 87 91 93 93 CONTENTS IX 5.1.7 The Curl 5.1.8 Orientability 5.2 The Divergence Theorem and Stokes' Theorem 5.3 Physical Interpretation of Divergence Thm 5.4 A Physical Interpretation of Stokes' Theorem 5.5 Proof of the Divergence Theorem 5.6 Sketch of a Proof for Stokes' Theorem 5.7 Books 5.8 Exercises 94 94 95 97 98 99 104 108 108 Differential Forms and Stokes' Thm 6.1 Volumes of Parallelepipeds 6.2 Diff Forms and the Exterior Derivative 6.2.1 Elementary k-forms 6.2.2 The Vector Space of k-forms 6.2.3 Rules for Manipulating k-forms 6.2.4 Differential k-forms and the Exterior Derivative 6.3 Differential Forms and Vector Fields 6.4 Manifolds 6.5 Tangent Spaces and Orientations 6.5.1 Tangent Spaces for Implicit and Parametric Manifolds 6.5.2 Tangent Spaces for Abstract Manifolds 6.5.3 Orientation of a Vector Space 6.5.4 Orientation of a Manifold and its Boundary 6.6 Integration on Manifolds 6.7 Stokes'Theorem 6.8 Books 6.9 Exercises 111 112 115 115 118 119 122 124 126 132 Curvature for Curves and Surfaces 7.1 Plane Curves 7.2 Space Curves 7.3 Surfaces 7.4 The Gauss-Bonnet Theorem 7.5 Books 7.6 Exercises 145 145 148 152 157 158 158 Geometry 8.1 Euclidean Geometry 8.2 Hyperbolic Geometry 8.3 Elliptic Geometry 8.4 Curvature 161 162 163 166 167 132 133 135 136 137 139 142 143 BIBLIOGRAPHY 333 [52] Halmos, Paul R, Finite-Dimensional Vector Spaces, Reprinting of the 1958 second edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1993 viii+200 pp [53] Halmos, Paul R Naive Set Theory, Reprinting of the 1960 edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1974 vii+104 pp [54] Halmos, Paul R., Measure Theory, Graduate Texts in Mathematics, 18, Springer-Verlag, New York, 1976,305 pp [55] Hartshorne, Robin, Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2000 xii+526 pp [56] David Henderson, Differential Geometry: A Geometric Introduction, Prentice Hall, 1998 250 pp [57] Herstein, I., Topics in Algebra, Second edition, John Wiley & Sons, 1975 [58] Hilbert, D and Cohn-Vossen, S., Geometry and the Imagination, AMS Chelsea, 1999 357 pp [59] Hill, Victor E., IV, Groups and Characters, Chapman and Hall/CRC, Boca Raton, FL, 1999.256 pp [60] Hintikka, Jaakko, The Principles of Mathematics Revisited, With an appendix by Gabriel Sandu, Cambridge University Press, Cambridge, 1998 302 pp [61] Hofstadter, Douglas R, Godel, Escher, Bach: An Eternal Golden Braid, Basic Books, Inc., Publishers, New York, 1979 777 pp [62] Howard, Paul and Rubin, Jean, Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, RI, 1998 viii+432 pp [63] Hubbard, Barbara Burke, The World According to Wavelets: The Story of a Mathematical Technique in the Making, Second edition, A K Peters, Ltd., Wellesley, MA, 1998 286 pp [64] Hubbard, J and Hubbard, B., Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Prentice Hall, 1999 687 pp [65] Hungerford, T., Algebra, Eighth edition, Graduate Texts in Mathematics, 73, Springer, 1997 502 pp 334 BIBLIOGRAPHY [66J Iserles, Arieh, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1996 396 pp [67J Jackson, Dunham, Fourier Series and Orthogonal Polynomials, Carus Monograph Series, no 6, Mathematical Association of America, Oberlin, Ohio, 1941 xii+234 pp [68J Jacobson, N., Basic Algebra, Vol I and II, Second edition, W.H Freeman, 1985 [69J John, Fritz, Partial Differential Equations, Reprint of the fourth edition Applied Mathematical Sciences, 1, Springer-Verlag, New York, 1991 x+249 pp [70J Jones, Frank, Lebesgue Integration on Euclidean Space, Revised edition, Jones and Bartlett Publishers, Boston, MA, 2001 608 pp [71J Kac, Mark, Statistical Independence in Probability, Analysis and Number Theory, The Carus Mathematical Monographs, No 12, Mathematical Association of America, New York 1969 xiv+93 pp [72J Kelley, John L., General Topology, Graduate Texts in Mathematics, 27 Springer-Verlag, New York-Berlin, 1975 xiv+298 pp [73J Kline, Morris, Mathematics and the Search for Knowledge, Oxford University Press, New York, 1972 1256 pp [74J Kobayashi, Shoshichi and Nomizu, Katsumi, Foundations of Differential Geometry, Vol I, Wiley Classics Library A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1996 xii+329 [75J Kobayashi, Shoshichi and Nomizu, Katsumi, Foundations of Differential Geometry, Vol II, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1996 xvi+468 pp [76J Kolmogorov, A N.and Fomin, S V., Introductory Real Analysis, Translated from the second Russian edition and edited by Richard A Silverman, Dover Publications, Inc., New York, 1975 xii+403 pp / [77J Krantz, Steven G., Function Theory of Several Complex Variables, Second edition, AMS Chelsea, 2001 564 pp [78J Krantz, Steven G., Complex Analysis: The Geometric Viewpoint, Carus Mathematical Monographs, 23, Mathematical Association of America, Washington, DC, 1990 210 pp BIBLIOGRAPHY 335 [79] Lang, Serge, Algebra, Third edition, Addison-Wesley, 1993 , 904 pp [80] Lang, Serge, Undergraduate Analysis, Second edition, Undergraduate Texts in Mathematics, Springer-Verlag, 1997,642 pp [81] Lang, Serge and Murrow, Gene, Geometry, Second edition, Springer-Verlag, 2000, 394 pp [82] Mac Lane, Saunders, Mathematics, Form and Function, Springer-Verlag, New York-Berlin, 1986 xi+476 pp [83] Marsden, Jerrold E and Hoffman, Michael J., Basic Complex Analysis, Third edition, W H Freeman and Company, New York, 1999 600 pp [84] McCleary, John, Geometry from a Differentiable Viewpoint, Cambridge University Press, Cambridge, 1995 320 pp [85] Millman, Richard and Parker, George D., Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, N J., 1977 xiv+265 pp [86] Morgan, Frank, Riemannian Geometry: A Beginner's Guide, Second edition, A K Peters, Ltd., Wellesley, MA, 1998 160 pp [87] Moschovakis, Yiannis N., Notes on Set Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1994 xiv+272 pp [88] Munkres, James R., Topology: A First Course, Second edition, PrenticeHall, Inc., Englewood Cliffs, N.J., 2000 537 pp [89] Nagel, Ernest and Newman, James R., Godel's Proof, New York University Press, New York 1960 ix+1l8 pp [90] Olver, P., Applications of Lie Groups to Diferential Equations, Second edition, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993 [91] O'Neill, Barrett, Elementary Differential Geometry, Second edition, Academic Press, New York-London 1997 448 pp [92] Palka, Bruce P., An Introduction to Complex Function Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991 xviii+559 pp [93] Polya, George, Tarjan, Robert E.and Woods, Donald R., Notes on Introductory Combinatorics, Progress in Computer Science, 4, Birkhauser Boston, Inc., Boston, Mass., 1990 v+192 pp 336 BIBLIOGRAPHY [94] Protter, Murray H and Morrey, Charles B., Jr A First Course in Real Analysis, Second edition Undergraduate Texts in Mathematics, SpringerVerlag, New York, 1991 xviii+534 pp [95] Royden, H L., Real Analysis, Third edition, Prentice-Hall, 1988.434 pp [96] Rudin, Walter Principles of Mathematical Analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Dsseldorf, 1976 x+342 pp [97] Rudin, Walter, Real and complex analysis, Third edition, McGraw-Hill Book Co., New York, 1986 xiv+416 pp [98] Seeley, Robert T., An Introduction to Fourier Series and Integrals, W A Benjamin, Inc., New York-Amsterdam 1966 x+104 pp [99] Simmons, George, Differential Equations With Applications and Historical Notes, McGraw-Hill Higher Education, 1991,640 pp [100] Smullyan, Raymond M., Gadel's Incompleteness Theorems, Oxford Logic Guides, 19, The Clarendon Press, Oxford University Press, New York, 1992 xvi+139 pp [101] Spiegel, M., Schaum's Outline of Complex Variables, McGraw-Hill, 1983 [102] Spivak, M., Calculus, Third edition, Publish or Perish, 1994 670 pp [103] Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Westview Press, 1971 160 pp [104] Spivak, Michael, A Comprehensive Introduction to Differential Geometry, Vol I-V Third edition, Publish or Perish, Inc., 1979 [105] Stanley, Richard P., Enumerative Combinatorics, Vol 1, With a foreword by Gian-Carlo Rota, Cambridge Studies in Advanced Mathematics, 49 Cambridge University Press, Cambridge, 1997 337 pp [106] Sternberg, S., Group Theory and Physics, Cambridge University Press, Cambridge, 1995.443 pp [107] Stewart, Ian, Galois theory, Second edition, Chapman and Hall, Ltd., London, 1990 xxx+202 pp [108] Stewart, J., Calculus, Brooks/Cole Pub Co, third edition, 1995, 1015 pp [109] Strang, G., Linear Algebra and its Applications, Third edition, Harcourt College, 1988 505 pp BIBLIOGRAPHY 337 [110] Strikwerda, John C Finite Difference Schemes and Partial Differential Equations, The Wadsworth and Brooks/Cole Mathematics Series, Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, CA, 1989 xii+386 pp [111] Thorpe, John A., Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1994 xiv+267 pp [112] Trefethen, Lloyd and Bau, David, III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997 xii+361 pp [113] van der Waerden, B L., Algebra, Vol 1, Based in part on lectures by E Artin and E Noether, Translated from the seventh German edition by Fred Blum and John R Schulenberger, Springer-Verlag, New York, 1991 xiv+265 pp [114] van der Waerden, B 1., Algebra, Vol 2, Based in part on lectures by E Artin and E Noether, Translated from the fifth German edition by John R Schulenberger, Springer-Verlag, New York, 1991 xii+284 pp [115] van Lint, J H and Wilson, R M., A Course in Combinatorics, Second edition, Cambridge University Press, Cambridge, 2001 550 pp [116] Zygmund, A., Trigonometric Series, Vol I, II, Reprinting of the 1968 version of the second edition with Volumes I and II bound together, Cambridge University Press, Cambridge-New York-Melbourne, 1988 768 pp Index oklO, 1],6 O(n), 114 L Spaces, 248 GL(n,R),214 R n ,2 Standard Topology, 66 Heine-Borel, 69 Sn,214 Abelian Group, 216 Absolute Convergence, 39 Adams, C., xvii Adjacency Matrix, 309 Ahlfors, L., 197 Aho, A., 324 Algebraic Closure, 224 Algorithms, 308 Big Notation, 308 Input, 308 NP,317 Output, 308 P=NP?,316 Parallel, 324 Polynomial Time, 316 Sorting, 314 Amplitudes, 245 Analytic Functions and Harmonic Functions, 270 as Limit, 171, 172 Cauchy Integral Formula, 185 Cauchy's Theorem, 182 Cauchy-Riemann Equations, 172, 175 Complex Derivative, 173 Conformal Maps, 172, 191 Definition, 191 Riemann Mapping Theorem, 194 Hartog's Theorem, 196 Integral Representations, 172, 179 Nonholomorphic Function, 173 Power Series, 172, 187, 188 Angle between Curves, 191 Arc Length Parametrization, 146, 148 Arnold, T., xvii Artin, E., 21, 228 Artin, M., 228 Atkinson, K., 324 Axiom of Choice, 201, 206 Algebraically Closed, 224 Hausdorff Maximal Principle, 212 Non-measurable Sets, 208 Statement, 207 Zorn's Lemma, 212 Barnard, Y., xvii Barrett, D., xvii Barschdorff, M., xvii Bartle, R., 43 Basis for Topology, 74 First Countable, 74 Neighborhood Basis, 74 Second Countable, 74 Basis of a Vector Space, Schauder Basis, 249 Bau, D., 324 Berberian, S., 43 Berenstein, C., 198 Bernoulli Trials, 294 Big Notation, 308 Binomial Coefficients, 287 Birkoff, G., 228 Bishop, E., 206 Bocher, M., 21 Bolyai, J., 164 Boundary, 86, 136 Induced Orientation, 136 Bounded Sets, 69 Boyce, W., 282 Bressoud, D., 43 Brouwer, L., 206 Brualdi, R., 305 Budar, N., xvii Burden, R., 324 INDEX Burger, E., xvii Cameron, P., 305 Canary, D., xvii Cantor Diagonalization, 204 Cantor Set, 212, 234 Measure Zero, 235 Uncountable, 235 Cardinality, 201 Cauchy Integral Formula, 185 Cauchy's Theorem, 182 Cauchy-Riemann Equations, 172, 175 Cederberg, J., 162, 169 Central Limit Theorem, 294 Chain Rule, 52 Characteristic Functions, 237 Characteristic Polynomial, 16 Similar Matrices, 17 Cheslack-Postava, T., xvii Chung, K., 305 Ciarlet, P., 324 Circle as Manifold, 126, 128, 130 Curvature, 145 Calculation, 148 Circuit Euler, 310 Hamiltonian, 312 of a Graph, 310 Closed Sets, 64 Cohen, P., 206, 208, 211 Cohn-Vossen, S., 168 Combinatorial Formulas (~), 287 Binomial Coefficients, 287 Examples, 286 Compact Sets, 64 Complexity Theory, see Algorithms Conditional Probability, 290 Conformal, 172, 191 Definition, 191 Riemann Mapping Theorem, 194 Conjugate, 173 Connected Spaces, 64 Continuous Functions Single-Variable Definition, 25 Example, 25 Vector-Valued Definition, 49 Via open sets, 64 Convolution, 255 Conway, J., 198 339 Cooper, B., xvii Cormen, T., 308, 324 Correll, B., xvii Cosets,216 Countability, 201 Cantor Diagonalization, 204 Cardinality n, 201 Countably Infinite, 201 Measure Zero, 233 Integers, 202 Polynomials, 211 Products of Sets, 203 Rational Numbers, 203 Uncountability of Real Numbers, 204 Uncountably Infinite, 201 Cantor Set, 235 Cowles, L., xvii Coxeter, H.S.M., 169 Crystals and Symmetries, 213 Curl, 94 as Exterior Derivative, 124 Curvature Circle, 145 Calculation, 148 Gauss-Bonnet, 157 Geometry, 167 Angles of Triangles, 168 Gaussian Curvature, 168 Plane, 145 Plane Curves Definition, 147 Varying Tangents, 147 Space Curves Binormal Vector, 150 Helix, 151 Principal Curvature, 149 Principal Normal Vector, 149 Torsion, 150 Sphere, 145 Straight Line, 145 Surfaces, 152 Gauss Map, 153 Gaussian Curvature, 154 Hessian, 153 Mean Curvature, 154 Principal Curvatures, 154 Curve, 87 Angle between Curves, 191 Arc Length Parametrization, 146, 148 Plane Curves and Curvature, 147 Cyclic Group, 217 340 Cylinder Curvature, 156 d'Alembert Formula and the Wave Equation, 279 Davis, H., 258, 279 Davis, P., 198 De Souza, P., xvi Dennison, K., xvii Determinant, Multilinear Function Definition, 10 Product of Eigenvalues, 18 Recursive Definition, Signed Volume Definition, 11 Volume of Parallelepiped, 113 Differentiable Functions Single-Variable Definition, 26 Example, 28 Vector-Valued Chain Rule, 52 Computations, 51 Definition, 50 Implicit Function Theorem, 58 Inverse Function Theorem, 53 Differential Equations, 261 Fourier Transform Methods, 256 Heat Equation, 256 Definition, 270 Derivation, 271 Steady-State Solutions, 267 Integrability Conditions, 279 Laplacian Dirichlet Problem, 266 Harmonic, 266 Mean Value Principle, 266 Lewy's Example, 281 Linear, 262 Homogeneous, 262 Nonlinear, 262 Ordinary Definition, 261 Existence and Uniquenss of Solutions, 264 Lipschitz, 264 Picard Iteration, 265 Partial Definition, 261 Separation of Variables, 267 Symmetries, Groups and Lie Theory, 213 Wave Equation Change of Variables, 277 INDEX d'Alembert Formula, 279 Definition, 274 Derivation, 274 Differential Forms, Ill, 115 k-forms, 118 Maniplliating, 119 Multilinear Maps, 118 Vector Space of, 118 Wedge Product, 121 Wedging, 119 O-forms and functions, 124 I-forms and Vector Fields, 124 2-forms and Vector Fields, 124 3-forms and functions, 124 and Vector Fields, 124 Definition, 122 Elementary k-forms, 115, 118 and Parallelepiped, 118 Elementary I-forms, 117 Elementary 2-forms, 115 Exterior Derivative O-forms, 122 Definition, 123 Integrating k-forms, 138 Dimension Vector Space, Diprima, R., 282 Dirichlet Problem, 266 Divergence, 93 as Exterior Derivative, 124 Divergence Theorem, 82, 96 Physical Interpretation, 97 Proof,99 Carmo, M., 158 Doggart, C., xvii Double Elliptic Geometry, 166 Model, 166 Dual Space, 20 and k-forms, 125 Dual Map, 20 Dugundji, J., 78 Dunne, E., xvii Edges of a Graph, 309 Edwards, H., 223, 228 Eigenvalue, 16 Eigenvector, 16 Elliptic Geometries, 166 Axiom, 163 Double, 166 Model,166 Single, 166 Model, 167 INDEX Empty Set, 206 Equivalence Classes, 328 Equivalence Problems, xix Physics, xxii Equivalence Relations, 327 Euclid, 161, 164 Euclidean Geometry, 161, 162 Fifth Postulate, 162 Line, 162 Playfair's Axiom, 163 Surface, 162 Euler Circuit, 310 Evans, L., 282 Expected Value, 291 Linearity, 291 Exterior Derivative, 111, 115 and Curl, 124 and Divergence, 124 and the Gradient, 124 Definition, 123 O-forms, 122 Faires, J., 324 Feller, W., 305 Feynmann, R., 108 Fields, 223 Algebraically Closed, 224 Field Extensions, 223 Fixed Fields, 225 Group of Automorphisms, 225 Normal,226 Splitting Fields, 225 Fifth Postulate, 162 Finney, R., 108 Fleming, W., 60, 142 Folland, G., 241, 258, 266, 282 Fomin, S., xvii Fourier Analysis Fourier Integral, 253 Fourier Series Amplitudes, 245 Convergence in the Mean, 251 Definition, 245, 246 Fourier Coefficients, 245 Gibb's Phenomenon, 251 Uniform Convergence, 252 Fourier Transform Solving Differential Equations, 256 Convolution, 255 Definition, 253 Properties, 254 Hilbert Spaces, see Hilbert Spaces 341 Fraleigh, J., 228 French, C., xvii Functions Analytic, see Analytic Functions as O-forms, 124 as 3-forms, 124 Characteristic, 237 Continuity via open sets, 64 Gradient, 93 Harmonic, 266 Holomorphic, see Analytic Functions Integral Representations of, 179 Lipschitz, 264 Measurable, 238 on Manifolds, 131 Periodic, 244 Random Variables, 289 Single-Variable Continuous, 25 Differentiable, 26 Limit existing, 23 Pointwise Convergence, 35 Uniform Convergence, 36 Weierstrass Example, 41 Step, 238 Vector-Valued Chain Rule, 52 Continuous, 49 Definition, 47 Differentiable, 50 Example, 48 Implicit Function Theorem, 58 Inverse Function Theorem, 53 Limit existing, 49 Fundamental Theorem of Calculus, 31, 81,95,224 Statement, 32 Gi:idel's Theorem, 210 Gi:idel, K., 208, 210, 307 Gallian, J., 228 Galois Theory, 223 Field Extensions, 223 Fixed Fields, 225 Fundamental Theorem, 226 Group of Automorphisms, 225 Normal,226 Quadratic Equation, 224 Splitting Fields, 225 Gans, D., 169 Garling, D., 224, 228 Gauss Map, 153, 159 342 Hessian, 153 Gauss, C., 164 Gauss-Bell Curve, 285, 295 Gauss-Bonnet Theorem, 157 Gaussian Curvature, 154 Geometries, 168 Gay, R., 198 Genus, 157 Geometry Curvature, 167 Angles of Triangle, 168 Gaussian, 168 Elliptic, 166 Axiom, 163 Double, 166 Model for Double, 166 Model for Single, 167 Single, 166 Euclidean, 161, 162 Fifth Postulate, 162 Line, 162 Playfair's Axiom, 163 Surface, 162 Hyperbolic, 163 Axiom, 164 Unit Disc, 169 Upper Half Plane, 164 Gibb's Phenomenon, 251 Goldstern, M., 211 Gradient, 93 as Exterior Derivative, 124 as Normal Vector, 153 Graham, R., 305 Graph Adjacency Matrix, 309 Circuit, 310 Connected, 325 Definition, 309 Edges, 309 Euler Circuit, 310 Examples, 309 Hamiltonian Circuit, 312 Konigsberg Bridge Problem, 310 Path, 310 Trees Binary, 314 Definition, 313 Leaves, 313 Root, 313 Sorting, 314 Vertices, 309 Gray, A., 158 Green's Theorem, 82, 179, 182 INDEX Greene, R., 197 Grimmett, G., 305 Group, 213 Abelian, 216 Cosets,216 Cyclic, 217 Definition, 213 Homomorphism, 216 Identity, 214 Inverse, 214 Kernel,218 Order, 218 Orthogonal, 114 Permutation, 120, 214 Flips, 120 Notation, 215 Shuffles, 121, 143 Sign of, 120 Transpositions, 120 Quotient Group, 217 Representation Theory, see Representation Theory Rotations, 214 Solvable, 227 Subgroup, 216 Normal, 217 Sylow's Theorem, 218 Halliday, D., 108 Halmos, P., 21, 211, 241 Hamiltonian Circuit, 312 Harmonic Analysis, see Fourier Analysis Harmonic Functions, 266 and Analytic Functions, 270 Hartog's Theorem, 196 Hartshorne, R., 169 Hausdorff, 64, 127 Hausdorff Maximal Principle, 212 Haynes, R., xvii Heat Equation, see Differential Equations Heine-Borel Theorem, 69 Helix, Curvature of, 151 Henderson, D., 158 Herstein, I., 219, 228 Hessian, 153 Eigenvalues as Principal Curvatures, 154 Gaussian Curvature, 154 Mean Curvature, 154 Hilbert Spaces Complete, 248 343 INDEX Definition, 248 Inner Product, 247 Orthogonal Vectors, 248 Schauder Basis, 249 Square-integrable Functions, 248 Hilbert, D., 168, 206 Hill, V., 228 Hinman, P., xvii Hintikka, J., 211 Hoffman, M., 197 Hofstadter, R., 211 Holomorphic, see Analytic Functions Homotopy, 184 Simply Connected, 194 Hopcroft, J., 324 Howard, P., 208 Hubbard, B., 60, 142, 258 Hubbard, J., 60, 142 Hungerford, T., 228 Hyperbolic Geometry, 163 Axiom, 164 Unit Disc, 169 Upper Half Plane, 164 Ideal in Ring, 222 Implicit Function Theorem, 56 Independence in Probability Theory, 290 Independence Results, 210 Completeness, 210 Consistency, 210 Inner Measure, 233 Integers as Rings, 221 Size of, 201, 202 Integrability Conditions, 279 Integral Representations Analytic Functions, 179 Integrals and Differential Forms, 138 Area, 29 Path or line, 88 Infinitesimal Arc Length, 89 Riemann, 28 Definition, 31 Lower Sums, 31 Upper Sums, 31 Surface, 91 Intermediate Value Theorem, 321 Inverse Function Theorem, 53 Iserles, A., 324 Jackson, D., 258 Jacobian Chain Rule, 52 Computation, 51 Definition, 50 Implicit Function Theorem, 58 Inverse Function Theorem, 53 Parametrized Manifolds, 84, 127 Jacobson, N., 228 James, D., xvii John, F., 282 Jones, F., 241 Judah, H., 211 Konigsberg Bridge Problem, 310 Koronya, A., xvii Karni, S., xvii Kastermans, B., xvii Kelley, J., 78 Kennedy, C., xvii Kernel,5 Group, 218 Klein, F., 164, 166 Kline, M., 161 Klodginski, E., xvii Knapp, F G., xvii Knuth, D., 305 Kobayashi, S., 158 Krantz, S., 197, 198 Kravitz, S., xvii Kronecker, L, 206 Lang, S., 43, 169, 228 Laplacian, see Differential Equations Lebesgue Dominating Convergence Theorem, 231, 239 Lebesgue Integration, see Measure Theory Lebesgue Measure, see Measure Theory Leiserson, C., 308, 324 Length of Complex Number, 174 Lenhart, W., xvii Lewy's Example, 281 Lewy, H., 282 Limit Single-Variable Definition, 23 Example, 24 Vector-Valued Definition, 49 Line Integral, 88 Infinitesimal Arc Length, 89 Linear Transformations, as Matrices, Definition, 344 Eigenvalue, 16 Eigenvector, 16 Image, Kernel,5 Lipschitz Functions, 264 Lobatchevsky, N., 164 Loop Closed,180 Counterclockwise, 180 Homotopic, 184 Simple, 180 Lower Sums, 31 Mobius Strip, 94 Mac Lane, S., xvi, 228 Manifolds, 111, 126 Abstract, 129 Tangent Spaces, 133 Boundary, 136 Induced Orientation, 136 Circle, 126, 128, 130 Curve, 87 Differentiable Functions, 131 Equivalence, 132 Implicit, 128 Gradients, 128 Normal Space, 133 Tangent Space, 133 Orient ability, 94, 135 Induced on Boundary, 136 Parametrized, 84, 128 Boundary, 86 Jacobian, 84, 127 Tangent Space, 133 Projective Space, 143 Simple, 99 Sphere, 127, 143 Surface, 91 Surface Integral, 91 Torus, 143 Transition Functions, 129 Marsden, J., 197 Matrices, GL(n,R),214 Adjacency, 309 as Linear Transformations, Characteristic Polynomial, 16 Determinant Product of Eigenvalues, 18 Eigenvalue, 16 Eigenvector, 16 Multiplication by Complex Numbers, 177 INDEX Similar, 14 Symmetric, 19 Transpose, 13 McLeary, J., 158, 169 Mean Curvature, 154 Mean Value Principle for Laplacian, 266 Mean Value Theorem, 322 Measurable Functions, 238 Measure Theory Inner Measure, 233 Lebesgue Integral Definition, 238 Lebesgue Dominating Convergence Theorem, 239 Measurable Functions, 238 Measurable Sets, 232 Non-measurable sets, 208 Outer Measure, 232 Metric Spaces, 73 Complete, 248 Miekisz, J., xvii Millman, R., 158 Mizner, R., xvii Moore, E., 77 Moore, R., 77 Morgan, F., xvii, 158 Morrey, C., 43 Moschovakis, Y., 211 Multilinear k-forms, 118 Wedge Product, 121 Determinant, 10 Munkres, J., 78 Murrow, G., 169 Nagel, E., 211 Newman, J" 211 Newton's Method, 317 Nirenberg, L., 282 Noether, E., 21, 228 Nomizu, K., 158 Non-measurable Sets, 208 Normal Distribution, 295 NP Algorithms, 317 Numerical Analysis Newton's Method, 317 O'Neil, B., 158 Olver, P., 263 Open Sets, 64 Ordinary Differential Equations, see Differential Equations Orientability, 94, 136 INDEX as an Equivalence Relation, 135 Induced on Boundary, 136 Mobius Strip, 94 Vector Spaces, 135 Orthogonal Group, 114 Outer Measure, 232 P=NP?,316 Palka, B., 197 Parallelepiped, 112 and k-forms, 118 Volume, 113 Parametrizing Map for Manifolds, 127 Parker, G., 158 Partial Differential Equations, see Differential Equations Partially Ordered Set, 212 Pascal's Triangle, 305 Patashnik, 0.,305 Path Connected Spaces, 65 Path Integral, 88, 179 Infinitesimal Arc Length, 89 Pedersen, L., xvii Periodic Functions, 244 Picard Iteration, 265 Pigeonhole Principle, 306 Playfair's Axiom, 163 Poincare, R., 164 Pointwise Convergence of Functions Definition, 35 Example, 35 Polar Coordinates, 298 Polya, G., 305 Polynomial Ring Structure, 221 Roots Approximating, 318 Galois Theory, 213 Multiplicity, 18 Polynomial Time, 316 Poset, 212 Power Series, see Analytic Functions Probability Theory, 287 Bernoulli Trials, 294 Central Limit Theorem Statement, 294 Conditional Probability, 290 Definition Probability, 288 Expected Value, 291 Linearity, 291 Gauss-Bell Curve, 285, 295 Independence, 290 345 Normal Distribution, 295 Probability Definition, 287 Random Variables, 289 Sample Space, 287, 288 Standard Deviation, 293 Variance, 292 Projective Space as Manifold, 143 Protter, M., 43 Quadratic Equation, 224 Quotient Group, 217 Random Variables, 289 Rational Numbers Measure Zero, 233 Size of, 203 Real Numbers Size of, 201 Uncountability, 204 Reflexivity, 327 Representation Theory Direct Sum Representation, 220 Irreducible Representation, 221 Representation Definition, 219 Resnick, R., 108 Riemann Integral, 28 Definition, 31 Riemann Mapping Theorem, 194 Ring, 213 Definition, 221 Example Integers, 221 Polynomials, 221 Ideal,222 Quotient Ring, 222 Zariski Topology, 75 Spec(R), 75 Rivest, R., 308, 324 Root, S., xvii Royden, R., 239, 241 Rubin, J., 208 Rudin, W., 43, 241 Russell, B., 207 Russell-Zermelo Paradox, 207 Sacceri, G., 164 Sample Space, 287, 288 Scalars, 2, Schauder Basis, 249 Schippers, E., xvii 346 Seeley, R., 258 Series of Functions Convergence, 38 Shuffles, 143 Permutation Group, 121 Silva, C., xvii Silva, J., xvi Similar Matrices, 14 Characteristic Polynomial, 17 Simmon, G., 282 Single Elliptic Geometry, 166 Model,167 Smullyan, R., 211 Solvable Group, 227 Sorting Algorithms, 314 Space Curves, see Curvature Spectral Theorem, 323 Sphere as Manifold, 127, 143 Curvature, 145 Calculation, 155 Spiegel, M., 197 Spivak, M., 40, 41, 43, 60, 142, 158 Standard Deviation, 293 Stanley, R., 305 Step Functions, 238 Sternberg, S., 221, 228 Stewart, 1., 228 Stewart, J., 108 Stirling's Formula, 300 Stirzaker, D., 305 Stokes' Theorem, 82, 111 Classical, 96 General Form, 139 Physical Interpretation, 98 Proof, 104 Special Cases Classical Stokes' Theorem, 96 Divergence Theorem, 82, 96 Fundamental Theorem of Calculus,95 Green's Theorem, 82, 179, 182 Strang, G., 21 Strikwerda, J., 324 Successor Set, 206 Surface, 91 Curvature, 152 Gauss Map, 153 Gauss-Bonnet, 157 Gaussian Curvature, 154 Hessian, 153 Mean Curvature, 154 Principal Curvatures, 154 INDEX Genus, 157 Normal Vectors, 153 Surface Integral, 91 Sylow's Theorem, 218 Symmetric Matrices, 19 Symmetry, see Group, 327 Tangent Spaces, 133 Tarjan, R., 305 Thomas, G., 108 Thorpe, J., 158 Topology Basis, 74 First Countable, 74 Neighborhood Basis, 74 Second Countable, 74, 127 Bounded Sets, 69 Closed Sets, 64 Compact Sets, 64 Connected, 64 Continuous Functions, 64 Definition, 63 Gauss Bonnet, 157 Hausdorff, 64, 127 Induced Topology, 64 Metric Spaces, 73 Complete, 248 Open Sets, 64 Path Connected, 65 Standard Topology on R n, 66 Heine-Borel, 69 Zariski,75 Torsion of Space Curves, 150 Torus as Manifold, 143 Transitivity, 327 Transpose, 13 Trees, see Graphs Trefethon, L., 324 Type Theory, 207 Ullman, J., 324 Uniform Convergence of Functions Continuity, 37 Definition, 36 Fourier Series, 252 Series of Functions, 38 Upper Half Plane, 164 Van der Waerden, B., 21, 228 van Lint, J., 305 Variance, 292 347 INDEX Vector Fields and Differential Forms, 124 as I-forms, 124 as 2-forms, 124 Continuous, 82 Curl, 94 Definition, 82 Differentiable, 82 Divergence, 93 Vector Space, R n ,2 Basis, Definition, Dimension, Dual Space, 20 Dual Map, 20 Hilbert Space, see Hilbert Spaces Image, Inner Product, 247 Orthogonality, 248 Kernel, Key Theorem, 12 Linear Independence, Span, Subspace, Vector-Valued Functions, see Functions Vertices of a Graph, 309 Volume of Parallelepiped, 113 Wave Equation, see Differential Equations Waves Examples, 243 Wedge Product, 121 Weierstrass M-test, 38 Application Weierstrass Example, 42 Example, 39 Statement, 39 Weierstrass, K., 23, 40 Westerland, C., xvii Whitehead, A., 207 Wilson, R., 305 Woods, D., 305 Zariski Topology, 75 Spec(R),75 Zermelo, E., 207 Zermelo-Fraenkel Axioms, 206 Zermelo-Russell Paradox, 207 Zorn's Lemma, 212 Zygmund, A., 258 ... when he spent the year at the University of Washington, and 2000-1, when he spent the year at the University of Michigan, Ann Arbor All the Mathematics You Missed But Need to Know for Graduate School... see the broad outline of mathematics and to fill in the gaps in their knowledge The author explains the basic points and a few key results of the most important undergraduate topics in mathematics, ... the many topics that beginning graduate students at the best graduate schools are assumed to know Since there is unfortunately far more that is needed to be known for graduate school and for research