A Brief Table of Integrals (An arbitrary constant may be added to each integral.) 1. Jxndx = _1_ x n+ 1 (n *- -I) n + 1 2. J± dx = Inlxl 3. J eX dx = eX 4. JaXdx= ~ Ina 5. Jsinxdx = -cosx 6. J cosxdx = sinx 7. Jtanxdx = -Inlcosxl 8. Jcotxdx = Inlsinxl 9. J sec x dx = Inlsec x + tan xl = Inltan( ix + j w)1 10. J cscxdx = Inlcscx - cotxl = Inltan ~xl 11. Jsin- 1 ~ dx = xsin- 1 ~ +~ (a> 0) a a 12. Jcos- 1 ~ dx = xcos- 1 ~ -~ (a> 0) a a 13. Jtan- 1 ~ dx = xtan- 1 ~ - ~ln(a2 + x 2) (a> 0) 14. J sin2mx dx = 2~ (mx - sin mx cos mx) 15. J cos2mxdx = 2~ (mx + sinmxcosmx) 16. J sec2x dx = tan x 17. J csc2xdx = -cotx . n d sinn-1xcosx + n-I J' n-2 d 18 · J sIn x x = - -- sm x x n n n d cosn-1xsinx + n-l J n-2 d 19 · J cos x x = -- cos x x n n tann-lx 20. Jtannx dx = n _ 1 - Jtan n- 2x dx (n *- 1) cotn-lx J 21. J cotnx dx = - n=-T - cotn- 2x dx (n *- 1) 22 n d tanxsecn- 2x n - 2 n-2 d · Jsecx X= n-l + n-l Jsec x x (n *- 1) n d cot x cscn- 2X n - 2 J n 2 d 23 · J cs C x x = - n _ 1 + n _ 1 csc - x x (n *- I) 24. J sinh x dx = cosh x 25. J cosh x dx = sinh x 26. Jtanhxdx=lnlcoshxl 27. J cothxdx = Inlsinhxl 28. Jsechxdx = tan-1(sinhx) This table is continued on the endpapers at the back. Derivatives d(au) du dcos-1u -} du 1.--=a- 19. dx dx dx ~dx d(u + v - w) du dv dw dtan-1u = __ du 2. dx = dx + dx - dx 20. dx 1 + u2 dx d(uv) dv du 3 -- =u- +v- dcot-1u -1 du 'dx dx dx 21. dx = 1 + u2 dx d(u/v) v(du/dx) - u(dv/dx) 4. -d-- = 2 du X V -uy-u-2---1- dx d(u n ) n-l du 5 --=nu - · dx dx dcsc-1u _ -1 du 23. -~- u~ dx V 6 d(u ) = vu v- 1 du +uv(lnu) dv 'dx dx dx 24. d sinh u = cosh u du dx dx U 7 d(e ) =eudu · dx dx 25 dcoshu = sinhu du · dx dx d(e au ) au du 8. ~ =ae dx 26 dtanh u = sech2u du · dx dx d U du 9. ~ =au(lna)- 27 d coth u = _ (csch2u) du dx dx · dx dx d(ln u) 1 du 10 --=-- 28. d u = - (sech u)(tanh u) · dx udx S~~h ~: 29. dc~~hu = -(cschu)(cothu) ~: d . d dsinh-1u 1 du 12 smu = cosu~ 30. dx · dx dx ~dx 13 dcosu = -sinu du dcosh-1u = 1 du · dx dx 31. dx dx ~ 14 dtanu = sec~ du · dx dx dtanh-1u 32. 1 du dx = 1 - u2 dx 15 dcotu = -csc2u du · dx dx dcoth-1u 1 du 33. 16 dsecu = tanusecu du dx = 1 - u2 dx · dx dx dsech-1u -1 34. du dx 17. d~;u = -(cotu)(cscu) ~: dx u~ d csch -IU -1 1 du 35. du dx ~dx dx lul~ Continued on overleaf. Undergraduate Texts in Mathematics Editors S. Axler F. W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Childs: A Concrete Introduction to Anglin: Mathematics: A Concise History Higher Algebra. Second edition. and Philosophy. Chung: Elementary Probability Theory Readings in Mathematics. with Stochastic Processes. Third Anglin/Lambek: The Heritage of edition. Thales. Cox/Little/O'Shea: Ideals, Varieties, Readings in Mathematics. and Algorithms. Second edition. Apostol: Introduction to Analytic Croom: Basic Concepts of Algebraic Number Theory. Second edition. Topology. Armstrong: Basic Topology. Curtis: Linear Algebra: An Introductory Armstrong: Groups and Symmetry. Approach. Fourth edition. Axler: Linear Algebra Done Right. Devlin: The Joy of Sets: Fundamentals Second edition. of Contemporary Set Theory. Beardon: Limits: A New Approach to Second edition. Real Analysis. Dixmier: General Topology. BakINewman: Complex Analysis. Driver: Why Math? Second edition. Ebbinghaus/FlumlThomas: BanchofflWermer: Linear Algebra Mathematical Logic. Second edition. Through Geometry. Second edition. Edgar: Measure, Topology, and Fractal Berberian: A First Course in Real Geometry. Analysis. Elaydi: An Introduction to Difference Bix: Conics and Cubics: A Equations. Second edition. Concrete Introduction to Algebraic Exner: An Accompaniment to Higher Curves. Mathematics. Bremaud: An Introduction to Exner: Inside Calculus. Probabilistic Modeling. Fine/Rosenberger: The Fundamental Bressoud: Factorization and Primality Theory of Algebra. Testing. Fischer: Intermediate Real Analysis. Bressoud: Second Year Calculus. Flanigan/Kazdan: Calculus Two: Linear Readings in Mathematics. and Nonlinear Functions. Second Brickman: Mathematical Introduction edition. to Linear Programming and Game Fleming: Functions of Several Variables. Theory. Second edition. Browder: Mathematical Analysis: Foulds: Combinatorial Optimization for An Introduction. Undergraduates. Buchmann: Introduction to Foulds: Optimization Techniques: An Cryptography. Introduction. Buskes/van Rooij: Topological Spaces: Franklin: Methods of Mathematical From Distance to Neighborhood. Economics. Callahan: The Geometry of Spacetime: Frazier: An Introduction to Wavelets An Introduction to Special and General Through Linear Algebra. Re1avitity. Gamelin: Complex Analysis. Carter/van Brunt: The Lebesgue­ Gordon: Discrete Probability. StieItjes Integral: A Practical HairerlWanner: Analysis by Its History. Introduction. Readings in Mathematics. Cederberg: A Course in Modem Halmos: Finite-Dimensional Vector Geometries. Second edition. Spaces. Second edition. (continued after index) Jerrold Marsden Alan Weinstein CALCULUS III Second Edition With 431 Figures Springer Jerrold Marsden Alan Weinstein California Institute of Technology Department of Mathematics Control and Dynamical Systems 107-81 University of California Pasadena, California 91125 Berkeley, California 94720 USA USA Editorial Board S. Axler F. W. Gehring K.A. Ribet Mathematics Department Mathematics Department Department of San Francisco State East Hall Mathematics University University of Michigan University of California San Francisco, CA 94132 Ann Arbor, MI 48109 at Berkeley USA USA Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 26-01 Cover photograph by Nancy Williams Marsden. Library of Congress Cataloging in Publication Data Marsden, Jerrold E. Calculus III. (Undergraduate texts in mathematics) Includes index. I. Calculus. II. Weinstein, Alan. II. Marsden, Jerrold E. Calculus. III. Title. IV. Title: Calculus three. V. Series. QA303.M3372 1984b 515 84-5479 Printed on acid-free paper Previous edition Calculus © 1980 by The Benjamin/Cummings Publishing Company. © 1985 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue. New York, New York 10010 U.S.A. Typeset by Computype, Inc., S1. Paul, Minnesota. 9 8 765 ISBN -13: 978-0-387-90985-1 e-ISBN -13: 978-1-4612-5028-9 DOl: 10.1007/978-1-4612-5028-9 A member ofBertelsmannSpringer Science+Business Media GmbH To Nancy and Margo Preface The goal of this text is to help students learn to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies . • The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelled almost exactly on the exam­ ples; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best studep,ts. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. • The exercises come in groups of two and often four similar ones. • Answers to odd-numbered exercises are available in the back of the book, and every other odd exercise (that is, Exercise 1, 5, 9, 13, ... ) has a complete solution in the student guide. Answers to even­ numbered exercises are not available to the student. Placement of Topics Teachers of calculus have their own pet arrangement of -topics and teaching devices. After trying various permutations, we have arrived at the present arrangement. Some highlights are the following. • Integration occurs early in Chapter 4; antidifferentiationand the f notation with motivation already appear in Chapter 2. vIII Preface • Trigonometric functions appear in the first semester in Chapter 5. • The chain rule occurs early in Chapter 2. We have chosen to use rate-of -change problems, square roots, and algebraic functions in con­ junction with the chain rule. Some instructors prefer to introduce sinx and cosx early to use with the chain rule, but this has the penalty of fragmenting the study of the trigonometric functions. We find the present arrangement to be smoother and easier for the students. • Limits are presented in Chapter 1 along with the derivative. However, while we do not try to hide the difficulties, technicalities involving epsilonics are deferred until Chapter 11. (Better or curious students can read this concurrently with Chapter 2.) Our view is that it is very important to teach students to differentiate, integrate, and solve calcu­ lus problems as quickly as possible, without getting delayed by the intricacies of limits.