All the Mathematics You Missed but Need to Know for Graduate School

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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


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 © 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 1 Thomas 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 .G372002 51D-dc21 2001037644 ISBN 0 521 79285 1 hardback ISBN 0 521 79707 1 paperback

Dedicated to the Memory

Robert Mizner

Preface On the Structure of Mathematics 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 . 1 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 . . . . xiii xix xxiii XXlll

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