Multivariable Calculus with Applications

488 Pages·2018·8.71 MB·English

by  Peter D. Lax

Undergraduate Texts in Mathematics
Peter D. Lax
Maria Shea Terrell
Multivariable
Calculus with
Applications
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series Editors:
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
Advisory Board:
Colin Adams, Williams College
David A. Cox, Amherst College
L. Craig Evans, University of California, Berkeley
Pamela Gorkin, Bucknell University
Roger E. Howe, Yale University
Michael Orrison, Harvey Mudd College
Lisette G. de Pillis, Harvey Mudd College
Jill Pipher, Brown University
Fadil Santosa, University of Minnesota
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelations among different aspectsofthesubject.Theyfeature examples that
illustrate key concepts as well as exercises that strengthen understanding.
More information about this series at http://www.springer.com/series/666
Peter D. Lax Maria Shea Terrell
(cid:129)
Multivariable Calculus
with Applications
123
PeterD.Lax Maria SheaTerrell
CourantInstitute of MathematicalSciences Department ofMathematics
NewYork University Cornell University
NewYork,NY Ithaca, NY
USA USA
ISSN 0172-6056 ISSN 2197-5604 (electronic)
Undergraduate Texts inMathematics
ISBN978-3-319-74072-0 ISBN978-3-319-74073-7 (eBook)
https://doi.org/10.1007/978-3-319-74073-7
LibraryofCongressControlNumber:2017963518
MathematicsSubjectClassification(2010): 93C35,0001,97xx
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Preface
Ourpurposeinwritingamultivariablecalculustexthasbeentohelpstudentslearn
that mathematics is the language in which scientific ideas can be precisely for-
mulated and that science is a source of mathematical ideas that profoundly shape
the development of mathematics.
In calculus, students are expected to acquire a number of problem-solving
techniques and to practice using them. Our goal is to prepare students to solve
problems in multivariable calculus and to encourage them to ask, Why does cal-
culus work? As a result throughout the text we offer explanations of all the
importanttheoremstohelpstudentsunderstandtheirmeaning.Ouraimistofoster
understanding.
Thetextisintendedforafirstcourseinmultivariablecalculus.Onlyknowledge
of single variable calculus is expected. In some explanations we refer to the fol-
lowingtheoremsofcalculusasdiscussedforexampleinCalculusWithApplications,
Peter D.Lax and Maria Shea Terrell,Springer 2014.
(cid:129) MonotoneConvergenceTheoremAboundedmonotonesequencehasalimit.
(cid:129) Greatest Lower Bound and Least Upper Bound Theorem A set of numbers
that is bounded below has a greatest lower bound. A set of numbers that is
bounded above has a least upper bound.
Chapters1and2introducetheconceptofvectorsinRnandfunctionsfromRnto
Rm.Chapters3through8showhowtheconceptsofderivativeandintegral,andthe
important theorems of single variable calculus are extended to partial derivatives
and multiple integrals, and to Stokes’ and the Divergence Theorems.
Todopartialderivativeswithoutshowinghowtheyareusedisfutile.Therefore
inChapter8weusevectorcalculustoderiveanddiscussseveralconservationlaws.
In Chapter 9 we present and discuss a number of physical theories using partial
differential equations. We quote a final passage from the book:
We observe, with some astonishment, that except for the symbols used, the
equationsformembranesinwhichtheelasticforcesaresobalancedthattheydonot
vibrate, and heat-conducting bodies in which the temperature is so balanced that it
does not change, are identical.
v
vi Preface
Thereisnophysicalreasonwhytheequilibriumofanelasticmembraneandthe
equilibriumofheatdistributionshouldbegovernedbythesameequation,butthey
are, and so
Their mathematical theory is the same.
This is what makes mathematics a universal tool in dealing with problems of
science.
We thank friends and colleagues who have given us encouragement, helpful
feedback, and comments on early drafts of the book, especially Louise Raphael of
Howard University and Laurent Saloff-Coste and Robert Strichartz of Cornell
University. We also thank Cornell students in Math 2220 who suggested ways to
improve the text. We especially thank Prabudhya Bhattacharyya for his careful
readingandcommentsonthetextwhilehewasanundergraduateMathematicsand
Physics major at Cornell University.
The book would not have been possible without the support and help of Bob
Terrell. We owe Bob more than we can say.
New York, USA Peter D. Lax
Ithaca, USA Maria Shea Terrell
Contents
1 Vectors and matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Two-dimensional vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The norm and dot product of vectors. . . . . . . . . . . . . . . . . . . . . 9
1.3 Bilinear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 n-dimensional vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Norm and dot product in n dimensions . . . . . . . . . . . . . . . . . . . 25
1.6 The determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 Signed volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.8 Linear functions and their representation by matrices . . . . . . . . . 48
1.9 Geometric applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Other coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1 Differentiable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 The tangent plane and partial derivatives . . . . . . . . . . . . . . . . . . 115
3.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.5 Divergence and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4 More about differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.1 Higher derivatives offunctions of several variables. . . . . . . . . . . 161
4.2 Extrema offunctions of two variables . . . . . . . . . . . . . . . . . . . . 165
4.3 Extrema offunctions of several variables . . . . . . . . . . . . . . . . . . 177
4.4 Extrema on level sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
vii
viii Contents
5 Applications to motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.1 Motion in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2 Planetary motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.1 Introduction to area, volume, and integral . . . . . . . . . . . . . . . . . 205
6.2 The integral of a continuous function of two variables . . . . . . . . 223
6.3 Double integrals as iterated single integrals . . . . . . . . . . . . . . . . 238
6.4 Change of variables in a double integral . . . . . . . . . . . . . . . . . . 245
6.5 Integration over unbounded sets . . . . . . . . . . . . . . . . . . . . . . . . 253
6.6 Triple and higher integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7 Line and surface integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.1 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.2 Conservative vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
7.3 Surfaces and surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 310
8 Divergence and Stokes’ Theorems and conservation laws. . . . . . . . 333
8.1 Green’s Theorem and the Divergence Theorem in R2. . . . . . . . . 333
8.2 The Divergence Theorem in R3. . . . . . . . . . . . . . . . . . . . . . . . . 346
8.3 Stokes’ Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.4 Conservation laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
8.5 Conservation laws and one-dimensional flows . . . . . . . . . . . . . . 375
9 Partial differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
9.1 Vibration of a string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
9.2 Vibration of a membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
9.3 The conduction of heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
9.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
9.5 The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Answers to selected problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 477
Chapter 1
Vectors and matrices
Abstract The mathematical description of aspects of the natural world requires a
collection of numbers. For example, a position on the surface of the earth is
described by two numbers, latitude and longitude. To specify a position above the
earth requires a third number, the altitude. To describe the state of a gas we have
to specify its density and temperature; if it is a mixture of gases like oxygen and
nitrogen, wehave tospecify their proportion. Such situations are abstracted inthe
conceptofavector.
1.1 Two-dimensionalvectors
Definition1.1.Anorderedpairofnumbersiscalledatwo-dimensionalvec-
tor.Wedenoteavectorbyacapitalletter
U=(u ,u ).
1 2
Thenumbersu andu arecalledthecomponentsofthevectorU.Thesetof
1 2
alltwo-dimensionalvectors,denotedR2,iscalledtwo-dimensionalspace.
Weintroducethefollowingalgebraicoperationsfortwo-dimensionalvectors
(a) ThemultipleofavectorU=(u ,u )byanumberc, cU,isdefinedasthevector
1 2
obtainedbymultiplyingeachcomponentofUbyc:
cU=(cu ,cu ). (1.1)
1 2
(b) ThesumofvectorsU=(u ,u )andV=(v ,v ), U+V,isdefinedbyaddingthe
1 2 1 2
correspondingcomponentsofUandV:
U+V=(u +v ,u +v ). (1.2)
1 1 2 2
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