Jerrold Marsden and Alan Weinstein

367 Pages·2006·15.92 MB·English

Jerrold Marsden and Alan Weinstein
Area of rectangle A = lw
circle A = nr2
triangle A =+bh
Surface Area of sphere A = 47ir2
cylinder A = 2nrh
Volume of box V = lwh
sphere V=4 nr3
cylinder V = 7ir2h
4
cone V = (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
(T 1 (T 1 1
cosO=sin --0 ,cscO=sec --8 ,cot8=tan( ; --0
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
2 tan 0
tan 28 =
(1 - tan2@)
Half-angle formulas
si. n2 -0 -- -1 -COSB or sin2@= 1 - ~ 0 ~ 2 8
2 2 2
COS2 -8 -- -1 + case or cos2@ = 1 + cos28
2 2 2
tan -8 = --s-i-n-8- - --1 --cos8 or tan8= 1 - cos 28
2 1fcos8 sin0 sin 26'
Product formulas
sin 8 sin + = -21 [cos(8 - +) - cos(8 + +)I
+ 1 + +
cos 8 cos = - [cos(8 +) cos(8 - +)I
2
+ 1. + +
sin 8 cos = [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 11 25 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 CoxILittlelO'Shea: Ideals, Varieties,
and Philosophy. and Algorithms. Second edition.
Readings in Mathematics. Croom: Basic Concepts of Algebraic
AnglinILambek: The Heritage of Topology.
Thales. Curtis: Linear Algebra: An Introductory
Readings in Mathematics. Approach. Fourth edition.
Apostol: Introduction to Analytic Devlin: The Joy of Sets: Fundamentals
Number Theory. Second edition. of Contemporary Set Theory.
Armstrong: Basic Topology. Second edition.
Armstrong: Groups and Symmetry. Dixmier: General Topology.
Axler: Linear Algebra Done Right. Driver: Why Math?
Second edition. Ebbinghaus/Flum/Thomas:
Beardon: Limits: A New Approach to Mathematical Logic. Second edition.
Real Analysis. Edgar: Measure, Topology, and Fractal
BaWNewman: Complex Analysis. Geometry.
Second edition. Elaydi: An Introduction to Difference
BanchoffNVermer: Linear Algebra Equations. Second edition.
Through Geometry. Second edition. Exner: An Accompaniment to Higher
Berberian: A First Course in Real Mathematics.
Analysis. Exner: Inside Calculus.
Bix: Conics and Cubics: A FineIRosenberger: The Fundamental
Concrete Introduction to Algebraic Theory of Algebra.
Curves. Fischer: Intermediate Real Analysis.
Brkmaud: An Introduction to FlaniganIKazdan: Calculus Two: Linear
Probabilistic Modeling. and Nonlinear Functions. Second
Bressoud: Factorization and Primality edition.
Testing. Fleming: Functions of Several Variables.
Bressoud: Second Year Calculus. Second edition.
Readings in Mathematics. Foulds: Combinatorial Optimization for
Brickman: Mathematical Introduction Undergraduates.
to Linear Programming and Game Foulds: Optimization Techniques: An
Theory. Introduction.
Browder: Mathematical Analysis: Franklin: Methods of Mathematical
An Introduction. Economics.
Buchmann: Introduction to Frazier: An Introduction to Wavelets
Cryptography. Through Linear Algebra.
Buskeslvan Rooij: Topological Spaces: Gamelin: Complex Analysis.
From Distance to Neighborhood. Gordon: Discrete Probability.
Callahan: The Geometry of Spacetime: HairerNVanner: Analysis by Its History.
An Introduction to Special and General Readings in Mathematics.
Relavitity. Halmos: Finite-Dimensional Vector
CarterIvan Brunt: The Lebesgue- Spaces. Second edition.
Stieltjes Integral: A Practical Halmos: Naive Set Theory.
Introduction. Hammerlin/Hoffmann: Numerical
Cederberg: A Course in Modern Mathematics.
Geometries. Second edition. Readings in Mathematics.
Childs: A Concrete Introduction to Harris/Hirst/Mossinghoff: Combinatorics
Higher Algebra. Second edition. and Graph Theory.
Chung: Elementary Probability Theory Hartshorne: Geometry: Euclid and
with Stochastic Processes. Third Beyond.
edition.
(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.)
I
5. sinxdx= -cosx
I
6. cos x dx = sin x
I
7. tanx dx = -1nlcosxl
+
9. (secx dx = lqlsecx tanxl
I
10. cscxdx=ln~cscx-cotxl
ll./sin'*ad x=xsin-'5+a -2 (a>O)
I
12. cos-'5dx=xcos-'5-J= (a>O)
a a
i
1.3. tan-'*dx=xtan-'5-aln(a2+x2) (a>O)
a a 2
I 1
14. sin2mx dx = -( mx - sin mx cos mx)
2m
1 +
15. cos2mxd x = -( mx sin mx cos mx)
2m
16. Isec2xd x = tanx
I
17. csc2xd x = -cot x
Isinn-zx
18. I sinnxdx= - sinn- 'X cos x + n - 1 dx
n n
jcosn
cosn' xs in x + n - 1 -zX dx
n n
I
20. tannxd x = -- jtann-2xdx (n#l)
n-1
21. /cotnxdx = - -cOtn-'x - (cot*-2x dx (n + 1)
n-1
jsecn-zX
i tan x secnP2x n - 2 dx
22. secnx dx = + (n f 1)
n-1 n-1
+-J
I CO~XCSC"-~nX-2 cscn-2Xdx
23. cscnxdx= - n-1 n-1 (n + 1)
24. f sinh x dx = cosh x
I
26. tanh x dx = lnlcosh xl
I
27. coth x dx = lnlsinh xl
I
28. sechx dx = tan- '(sinhx )
This table is continued on the endpapers at the back.
Copyright 1985 Springer-Verlag. All rights reserved.
Derivatives
20. -d tan-'u - -+1 -d u
dx 1 u2 dx
23. -dcsc-'u -- -1 -du
dx uJ- du
24. dsinhu = cash u -du
dx dx
25. dcoshu = sinhu-d u
dx dx
dtanhu dU
26. = seCh2u
dx dx
27. dcothu = - (csch2u>-d u
dx dx
28. -Seth -- - (sech u)(tanhu )- du
dx dx
29. dcschu = -(cschu)(coth u)- du
dx dx
*
du
16. = tanusecu-
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 (*)a nd 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
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; antidijferentiation and the J
@
notation with motivation already appear in Chapter 2.
Copyright 1985 Springer-Verlag. All rights reserved.