Table Of ContentMarsden-3620111 VC˙FM September27,2011 9:49 i
SIXTH EDITION
Jerrold E. Marsden
California Institute of Technology, Pasadena
Anthony Tromba
University of California, Santa Cruz
W. H. Freeman and Company · New York
Marsden-3620111 VC˙FM September27,2011 9:49 ii
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Politicsisforthemoment.
Anequationisforeternity.
A.EINSTEIN
Somecalculustricksarequiteeasy.
Someareenormouslydifficult.Thefools
whowritethetextbooksof
advancedmathematicsseldomtakethetrouble
toshowyouhoweasytheeasycalculationsare.
SILVANUSP.THOMPSON,CALCULUSMADEEASY,MACMILLAN(1910)
LibraryofCongressControlNumber:2011931725
ISBN-13:978-1-4292-1508-4
ISBN-10:1-4292-1508-9
(cid:2)c2012,2003,1996,1988,1981,1976byW.H.FreemanandCompany
Allrightsreserved.
PrintedintheUnitedStatesofAmerica
Firstprinting
W.H.FreemanandCompanyPublishers
41MadisonAvenue
NewYork,NY10010
Houndmills,BasingstokeRG216XS,England
www.whfreeman.com
Marsden-3620111 VC˙FM September27,2011 9:49 iii
Contents
Preface ix
Acknowledgements xi
Historical Introduction: A Brief Account xiii
Prerequisites and Notation xxiii
1 The Geometry of Euclidean Space 1
1.1 VectorsinTwo-andThree-DimensionalSpace 1
1.2 TheInnerProduct,Length,andDistance 19
1.3 Matrices,Determinants,andtheCrossProduct 31
1.4 CylindricalandSphericalCoordinates 52
1.5 n-DimensionalEuclideanSpace 60
ReviewExercisesforChapter1 70
2 Differentiation 75
2.1 TheGeometryofReal-ValuedFunctions 76
2.2 LimitsandContinuity 88
2.3 Differentiation 105
2.4 IntroductiontoPathsandCurves 116
2.5 PropertiesoftheDerivative 124
2.6 GradientsandDirectionalDerivatives 135
ReviewExercisesforChapter2 144
3 Higher-Order Derivatives:
Maxima and Minima 149
3.1 IteratedPartialDerivatives 150
3.2 Taylor’sTheorem 158
3.3 ExtremaofReal-ValuedFunctions 166
3.4 ConstrainedExtremaandLagrangeMultipliers 185
3.5 TheImplicitFunctionTheorem[Optional] 203
ReviewExercisesforChapter3 211
iii
Marsden-3620111 VC˙FM September27,2011 9:49 iv
iv Contents
4 Vector-Valued Functions 217
4.1 AccelerationandNewton’sSecondLaw 217
4.2 ArcLength 228
4.3 VectorFields 236
4.4 DivergenceandCurl 245
ReviewExercisesforChapter4 260
5 Double and Triple Integrals 263
5.1 Introduction 263
5.2 TheDoubleIntegralOveraRectangle 271
5.3 TheDoubleIntegralOverMoreGeneralRegions 283
5.4 ChangingtheOrderofIntegration 289
5.5 TheTripleIntegral 294
ReviewExercisesforChapter5 304
6 The Change of Variables Formula and
Applications of Integration 307
6.1 TheGeometryofMapsfromR2 toR2 308
6.2 TheChangeofVariablesTheorem 314
6.3 Applications 329
6.4 ImproperIntegrals[Optional] 339
ReviewExercisesforChapter6 347
7 Integrals Over Paths and Surfaces 351
7.1 ThePathIntegral 351
7.2 LineIntegrals 358
7.3 ParametrizedSurfaces 375
7.4 AreaofaSurface 383
7.5 IntegralsofScalarFunctionsOverSurfaces 393
7.6 SurfaceIntegralsofVectorFields 400
7.7 ApplicationstoDifferentialGeometry,Physics,
andFormsofLife 413
ReviewExercisesforChapter7 423
Marsden-3620111 VC˙FM September27,2011 9:49 v
Contents v
8 The Integral Theorems of Vector Analysis 427
8.1 Green’sTheorem 428
8.2 Stokes’Theorem 439
8.3 ConservativeFields 453
8.4 Gauss’Theorem 461
8.5 DifferentialForms 476
ReviewExercisesforChapter8 490
Answers to Odd-Numbered Exercises 493
Index 533
Photo Credits 545
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Marsden-3620111 VC˙FM September27,2011 9:49 vii
To Jerrold E. Marsden, 1942--2010
JerryMarsden,CarlF.BraundistinguishedProfessoratthe
CaliforniaInstituteofTechnology,FellowoftheRoyalSociety
(aswasIsaacNewton),andoneoftheworld’spre-eminent
appliedmathematicians,passedawayonSeptember21,2010,
whileworkingonthesixtheditionofVectorCalculus.Jerry’s
interestswereunusuallybroad;hisworkinfluencedphysicists,
engineers,lifescientists,andmathematiciansacrossthescientific
andengineeringspectrum.Inadditiontohismanypublications
(over400archivalandconferencepapersand21books)and
majorscientificprizes,hewasabrilliantexpositorandteacher.
Hemotivatedandencouragedcolleaguesandstudentsalike,
aroundtheworldandacrossanastonishingarrayofdisciplines.
Hewasawonderfulpersonandaclosefriendforalmosthalfa
century.Hewillbesorelymissed.
—AnthonyTromba
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Marsden-3620111 VC˙FM September27,2011 9:49 ix
Preface
Thistextisintendedforaone-semestercourseinthecalculusoffunctionsofseveral
variablesandvectoranalysis,whichisnormallytaughtatthesophomorelevel.Inaddi-
tiontomakingchangesandimprovementsthroughoutthetext,wehavealsoattempted
toconveyasenseofexcitement,relevance,andimportanceofthesubjectmatter.
Prerequisites
Sometimescoursesinvectorcalculusareprecededbyafirstcourseinlinearalgebra,
butthisisnotanessentialprerequisite.Werequireonlythebarerudimentsofmatrix
algebra,andthenecessaryconceptsaredevelopedinthetext.Ifthiscourseispreceded
byacourseinlinearalgebra,theinstructorwillhavenodifficultyenhancingthematerial.
However,wedoassumeaknowledgeofthefundamentalsofone-variablecalculus—the
processofdifferentiationandintegrationandtheirgeometricandphysicalmeaningas
wellasaknowledgeofthestandardfunctions,suchasthetrigonometricandexponential
functions.
The Role of Theory
The text includes much of the basic theory as well as many concrete examples and
problems. Some of the technical proofs for theorems in Chapters 2 and 5 are given
in optional sections that are readily available on the Book Companion Web Site at
www.whfreeman.com/marsdenvc6e(seethedescriptiononthenextpage).Section2.2,
onlimitsandcontinuity,isdesignedtobetreatedlightlyandisdeliberatelybrief.More
sophisticatedtheoreticaltopics,suchascompactnessanddelicateproofsinintegration
theory,havebeenomitted,becausetheyusuallybelongtoamoreadvancedcoursein
realanalysis.
Concrete and Student-Oriented
Computational skills and intuitive understanding are important at this level, and we
have tried to meet this need by making the book concrete and student-oriented. For
example, although we formulate the definition of the derivative correctly, it is done
byusingmatricesofpartialderivativesratherthanabstractlineartransformations.We
also include a number of physical illustrations such as fluid mechanics, gravitation,
andelectromagnetictheory,andfromeconomicsaswell,althoughknowledgeofthese
subjectsisnotassumed.
Order of Topics
Aspecialfeatureofthetextistheearlyintroductionofvectorfields,divergence,andcurl
inChapter4,beforeintegration.Vectoranalysisoftensuffersinacourseofthistype,
andthepresentarrangementisdesignedtooffsetthistendency.Togoevenfurther,one
mightconsiderteachingChapter3(Taylor’stheorems,maximaandminima,Lagrange
multipliers)afterChapter8(theintegraltheoremsofvectoranalysis).
New to This Edition
Thissixtheditionwascompletelyredesigned,butretainsandimprovesonthebalance
betweentheory,applications,optionalmaterial,andhistoricalnotesthatwaspresentin
earliereditions.
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