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Jerrold Marsden and Alan Weinstein

Area of rectangle

circle triangle

Surface Area of sphere

cylinder

Volume of box

sphere cylinder cone A = lw A = nr2

A =+bh

A = 47ir2

A = 2nrh

V = lwh

V= 4nr3

V = 7ir2h

V = 4 (area of base) x (height)

Trigonometric Identities

Pythagorean

cos28 + sin26' = I, 1 + tan28 = sec28, cot2@ + 1 = csc28

Parity

sin(-8) = -sinO, cos(-8) = cos8, tan(-8) = -tan0

CSC(-8) = -cscB, sec(-8) = sec8, cot(-%) = -cot8

Co-relations

cosO=sin --0 ,cscO=sec --8 ,cot8=tan --0 (T 1 (T 1 (; 1

Addition formulas

sin(@ + 4) = sin 8 cos + + cos 0 sin + sin(8 - +) = sin 8 cos + - cos 0 sin + cos(8 + +) = cos 0 cos + - sin 8 sin + cos(8 - +) = cos 8 cos + + sin 8 sin + (tan 0 + tan +) tan(@ + +) = (1 - tan 8 tan +) (tan 8 - tan +) tan(8 - +) = (1 + tan 8 tan +)

Double-angle formulas

sin 28 = 2 sin 8 cos 8 cos 28 = cos28 - sin28 = 2 cos28 - I = 1 - 2 sin28 tan 28 =

2 tan 0

(1 - tan2@)

Half-angle formulas

. 2 0 - 1 -COSB or sin2@ =

1 - ~0~28

sin - - -

2 2 2

2 8 - 1 + case or cos2@ =

COS - - -

1 + cos28

2 2 2

8 sin8

-1-cos8 or tan8= tan - = ------- - -

1 - cos 28

2

1fcos8 sin0 sin 26'

Product formulas

1 sin 8 sin + = - [cos(8 - +) - cos(8 + +)I 2 1 cos

8 cos + = - [cos(8 + +) + cos(8 - +)I

2 sin

8 cos + = 1. [sin(@ + +) + sin(8 - +)I

2 Copyright 1985 Springer-Verlag. All rights reserved.

To Nancy and Margo

Copyright 1985 Springer-Verlag. All rights reserved.

Jerrold Marsden Alan Weinstein

California Institute of Technology Department of Mathematics Control and Dynamical Systems 107-81 University of California

Pasadena, California 9 1

125 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

Subiect Classification (2000): 26-01

Cover photograph by Nancy Williams

Marsden.

Library of Congress Cataloging in Publication Data

Marsden, Jerrold E.

Calculus 11.

(Undergraduate texts in mathematics)

Includes index.

1. Calculus.

11. Weinstein, Alan.

11. Marsden, Jerrold E. Calculus. 111. Title.

IV. Title: Calculus two. V. Series.

QA303.M3372 1984b 5 15 84-5480

Previous edition

Calculus O 1980 by The Benjamin/Cummings Publishing Company.

O 1985 by Springer-Veriag New York Inc.

All rights resewed. 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., St. Paul, Minnesota.

Printed and bound by

Sheridan Books, Inc., Ann Arbor, MI.

Printed

in the United States of America.

ISBN 0-387-90975-3

ISBN 3-540-90975-3 SPIN 10792714

Springer-Verlag New York Berlin Heidelberg

A member of BerteIsmannSpringer Science+Business Media GmbH Copyright 1985 Springer-Verlag. All rights reserved.

Second Edition

With 297 Figures

Springer

Copyright 1985 Springer-Verlag. All rights reserved.

Undergraduate Texts in Mathematics

Anglin: Mathematics: A Concise History

and Philosophy.

Readings in Mathematics.

AnglinILambek: The Heritage of

Thales.

Readings in Mathematics.

Apostol: Introduction to Analytic

Number Theory. Second edition.

Armstrong: Basic Topology.

Armstrong: Groups and Symmetry.

Axler: Linear Algebra Done Right.

Second edition.

Beardon: Limits: A New Approach to

Real Analysis.

BaWNewman: Complex Analysis.

Second edition.

BanchoffNVermer: Linear Algebra

Through Geometry. Second edition.

Berberian: A First Course in Real

Analysis.

Bix: Conics and Cubics: A

Concrete Introduction to Algebraic

Curves.

Brkmaud: An Introduction to

Probabilistic Modeling.

Bressoud: Factorization and Primality

Testing.

Bressoud: Second Year Calculus.

Readings in Mathematics.

Brickman: Mathematical Introduction

to Linear Programming and Game

Theory.

Browder: Mathematical Analysis:

An Introduction.

Buchmann: Introduction to

Cryptography.

Buskeslvan Rooij: Topological Spaces:

From Distance to Neighborhood.

Callahan: The Geometry of Spacetime:

An Introduction to Special and General

Relavitity.

CarterIvan Brunt: The Lebesgue-

Stieltjes Integral: A Practical

Introduction.

Cederberg: A Course in Modern

Geometries. Second edition.

Childs: A Concrete Introduction to

Higher Algebra. Second edition.

Chung: Elementary Probability Theory

with Stochastic Processes. Third edition.

CoxILittlelO'Shea: Ideals, Varieties,

and Algorithms. Second edition.

Croom: Basic Concepts of Algebraic

Topology.

Curtis: Linear Algebra: An Introductory

Approach. Fourth edition.

Devlin: The Joy of Sets: Fundamentals

of Contemporary Set Theory.

Second edition.

Dixmier: General Topology.

Driver: Why Math?

Ebbinghaus/Flum/Thomas:

Mathematical Logic. Second edition.

Edgar: Measure, Topology, and Fractal

Geometry.

Elaydi: An Introduction to Difference

Equations. Second edition.

Exner: An Accompaniment to Higher

Mathematics.

Exner: Inside Calculus.

FineIRosenberger: The Fundamental

Theory of Algebra.

Fischer: Intermediate Real Analysis.

FlaniganIKazdan: Calculus Two: Linear

and Nonlinear Functions. Second edition.

Fleming: Functions of Several Variables.

Second edition.

Foulds: Combinatorial Optimization for

Undergraduates.

Foulds: Optimization Techniques: An

Introduction.

Franklin: Methods of Mathematical

Economics.

Frazier: An Introduction to Wavelets

Through Linear Algebra.

Gamelin: Complex Analysis.

Gordon: Discrete Probability.

HairerNVanner: Analysis by Its History.

Readings in Mathematics.

Halmos: Finite-Dimensional Vector

Spaces. Second edition.

Halmos: Naive Set Theory.

Hammerlin/Hoffmann: Numerical

Mathematics.

Readings in Mathematics.

Harris/Hirst/Mossinghoff: Combinatorics

and Graph Theory.

Hartshorne: Geometry: Euclid and

Beyond.

(continued ajier index) Copyright 1985 Springer-Verlag. All rights reserved.

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 Copyright 1985 Springer-Verlag. All rights reserved.

A Brief Table of Integrals

(An arbitrary constant may.be added to each integral.)

5. sinxdx= -cosx

I

6. cos x dx = sin x

I

7. tanx dx = -1nlcosxl

I

9. (secx dx = lqlsecx + tanxl

10. cscxdx=ln~cscx-cotxl

I ll./sin'*dx=xsin-'5+-2 a a (a>O)

12. cos-'5dx=xcos-'5-J= (a>O)

I a a

1.3. tan-'*dx=xtan-'5-aln(a2+x2) (a>O)

i a a 2 I 1

14. sin2mx dx = - (mx - sin mx cos mx)

2m 1

15. cos2mx dx = - (mx + sin mx cos mx)

2m

16. Isec2x dx = tanx

17. csc2x dx = -cot x I I sinn- 'X cos x + n - 1 Isinn-zx dx

18. sinnxdx= -

n n cosn 'x sin x + n - 1 jcosn -zX dx n n I -jtann-2xdx (n#l) 20. tannx dx = - n-1 cOtn-'x - (cot*-2x dx (n + 1)

21. /cotnxdx = - -

n-1

22. secnx dx =

i tan x secnP2x + n - 2 jsecn-zX dx n-1 n-1 (n f 1) I

CO~XCSC"-~X n-2 cscn-2Xdx

23. cscnxdx= -

n-1 +-J n-1 (n + 1)

24. f sinh x dx = cosh x

26.
tanh x dx = lnlcosh xl I

27. coth x dx = lnlsinh xl

I

28. sechx dx = tan- '(sinh x)

I This table is continued on the endpapers at the back. Copyright 1985 Springer-Verlag. All rights reserved.

Derivatives

du

16. * = tanusecu- dx dx d tan-'u

1 du

20. - - - -

dx 1 + u2 dx dcsc-'u -

23. - -

-1 du dx uJ- du du

24. dsinhu = cash u -

dx dx du

25. dcoshu = sinhu-

dx dx

26. dtanhu = seCh2u dU

dx dx du

27. dcothu = - (csch2u> -

dx dx

28. - -

du Seth - - (sech u)(tanh u) - dx dx du

29. dcschu = -(cschu)(coth u) -

dx dx

34. d sech- 'u -

-1 du dx ulp'77 dx

Continued on overleaf

Copyright 1985 Springer-Verlag. All rights reserved.

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. a+ 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 students. 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

quotesdbs_dbs22.pdfusesText_28

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