Multivariable Calculus with MATLAB® : With Applications to Geometry and Physics

280 Pages·2017·8.13 MB·English

by  Ronald L. Lipsman

Ronald L. Lipsman · Jonathan M. Rosenberg
Multivariable
Calculus
with MATLAB®
With Applications to Geometry
and Physics
®
Multivariable Calculus with MATLAB
·
Ronald L. Lipsman Jonathan M. Rosenberg
Multivariable Calculus
®
with MATLAB
With Applications to Geometry and Physics
RonaldL.Lipsman JonathanM.Rosenberg
DepartmentofMathematics DepartmentofMathematics
UniversityofMaryland UniversityofMaryland
CollegePark,MD,USA CollegePark,MD,USA
ISBN978-3-319-65069-2 ISBN978-3-319-65070-8(eBook)
DOI10.1007/978-3-319-65070-8
LibraryofCongressControlNumber:2017949120
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Preface
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The preface of a book gives the authors their best chance to answer an extremely
importantquestion:Whatmakesthisbookspecial?
ThisbookisareworkingandupdatingforMATLABofourpreviousbook(joint
®
with Kevin R. Coombes) Multivariable Calculus with Mathematica ,
Springer,1998.Itrepresentsourattempttoenrichandenliventheteachingofmul-
tivariable calculus and mathematical methods courses for scientistsand engineers.
Mostbooksinthesesubjectsarenotsubstantiallydifferentfromthoseoffiftyyears
ago. (Well, they may include fancier graphics and omit several topics, but those
are minor changes.) This book is different. We do touch on most of the classical
topics; however, we have made a particular effort to illustrate each point with a
significant example. More importantly, we have tried to bring fundamental physi-
calapplications—Kepler’slaws,electromagnetism,fluidflow,energyestimation—
back to a prominent position in the subject. From one perspective, the subject of
multivariable calculus only existsbecause itcan beapplied toimportantproblems
inscience.
Inaddition,wehaveincludedadiscussionofthegeometricinvariantsofcurves
andsurfaces,providing,ineffect,abriefintroductiontodifferentialgeometry.This
materialprovidesanaturalextensiontothetraditionalsyllabus.
We believe that we have succeeded in resurrecting material that used to be in
the course while introducing new material. A major reason for that success is that
weusethecomputationalpowerofthemathematicalsoftwaresystemMATLABto
carry a large share of the load. MATLAB is tightly integrated into every portion
of this book. We use its graphical capabilities to draw pictures of curves and sur-
faces;weuse itssymbolical capabilities tocompute curvature and torsion;we use
itsnumericalcapabilitiestotackleproblemsthatarewellbeyondthetypicalmun-
daneexamplesoftextbooksthattreatthesubjectwithoutusingacomputer.Finally,
and this is something not done in any other books at this level, we give a serious
yet elementary explanation of how various numerical algorithms work, and what
their advantages and disadvantages are. Again, this is something that could not be
accomplishedwithoutasoftwarepackagesuchasMATLAB.
v
vi Preface
AsanadditionalbenefitfromintroducingMATLAB,weareabletoimprovestu-
dents’understandingofimportantelementsofthetraditionalsyllabus.Ourstudents
arebetterabletovisualizeregionsintheplaneandinspace.Theydevelopabetter
feelforthegeometricmeaningofthegradient;forthemethodofsteepestdescent;
fortheorthogonalityoflevelcurvesandgradientflows.Becausetheyhavetoolsfor
visualizingcrosssectionsofsolids,theyarebetterabletofindthelimitsofintegra-
tioninmultipleintegrals.
Tosummarize,wethinkthisbookisspecialbecause,byusingit:
(cid:129) studentsobtainabetterunderstandingofthetraditionalmaterial;
(cid:129) studentsseethedeepconnectionsbetweenmathematicsandscience;
(cid:129) studentslearnmoreabouttheintrinsicgeometryofcurvesandsurfaces;
(cid:129) studentsacquireskillusingMATLAB,apowerfulpieceofmodernmathematical
software;
(cid:129) instructorscanchoosefromamoreexcitingvarietyofproblemsthaninstandard
textbooks;and
(cid:129) both students and instructors are exposed to a more holistic approach to the
subject—onethatembracesnotonlyalgebraic/calculus-basedsolutionstoprob-
lems, but also numerical, graphical/geometric and qualitative approaches to the
subjectanditsproblems.
Conventions
Throughout the book, MATLAB commands, such as solve, are printed in type-
writer boldface. Theorems and general principles, such as: derivativesmeasure
change, are printed in a slanted font. When new terms, such as torsion, are intro-
duced,theyareprintedinanitalicfont.FilenamesandURLs(webaddresses)are
printedintypewriter font.Everythingelseisprintedinastandardfont.
At the start of each chapter, below the title, is a small illustration. Each is a
graphic generated by a MATLAB command. Most are taken from the MATLAB
solutiontooneoftheproblemsintheaccompanyingproblemset.Afewaretaken
fromthechapteritself.Finally,inthisPreface,thegraphicrepresentsamoreeclectic
choice.Weleaveittotheindustriousreadertoidentifythesourceofthesegraphics,
aswellastoreproducethefigure.
Acknowledgments
We above all want to thank our former collaborators for their contributions to this
project.KevinCoombes(nowattheDepartmentofBiomedicalInformaticsatOhio
®
State University) was a co-author of Multivariable Calculus with Mathematica
and kindly agreed to let us adapt that book for MATLAB. Brian Hunt was a
co-authorofAGuidetoMATLABandtaughtusmanyusefulMATLABtricksand
Preface vii
tips.PaulGreenhelpeddevelopMATLABexercisesformultivariablecalculusthat
eventuallyworkedtheirwayintothisbook.
JonathanRosenbergthankstheNationalScienceFoundationforitssupportunder
grantDMS-1607162.Anyopinions,findings,andconclusionsorrecommendations
expressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflectthe
viewsoftheNationalScienceFoundation.
CollegePark,MD,USA
RonaldL.Lipsman
October1,2017
JonathanM.Rosenberg
Contents
Preface........................................................... v
1 Introduction................................................... 1
1.1 BenefitsofMathematicalSoftware ............................ 2
1.2 What’sinThisBook ........................................ 3
1.2.1 ChapterDescriptions ................................. 3
1.3 What’sNotinThisBook .................................... 5
1.4 HowtoUseThisBook ...................................... 6
1.5 TheMATLABInterface ..................................... 7
1.5.1 AWordonTerminology .............................. 8
1.6 SoftwareVersions .......................................... 8
ProblemSetA.ReviewofOne-VariableCalculus .................... 9
GlossaryofMATLABCommands ................................. 12
OptionstoMATLABCommands .................................. 13
References..................................................... 13
2 VectorsandGraphics........................................... 15
2.1 Vectors ................................................... 15
2.1.1 ApplicationsofVectors ............................... 17
2.2 ParametricCurves .......................................... 19
2.3 GraphingSurfaces.......................................... 23
2.4 ParametricSurfaces......................................... 25
ProblemSetB.VectorsandGraphics............................... 27
GlossaryofMATLABCommands ................................. 31
OptionstoMATLABCommands .................................. 31
3 GeometryofCurves ............................................ 33
3.1 ParametricCurves .......................................... 33
3.2 GeometricInvariants........................................ 36
3.2.1 Arclength........................................... 36
3.2.2 TheFrenetFrame .................................... 37
ix
x Contents
3.2.3 CurvatureandTorsion ................................ 39
3.3 DifferentialGeometryofCurves .............................. 42
3.3.1 TheOsculatingCircle ................................ 42
3.3.2 PlaneCurves........................................ 43
3.3.3 SphericalCurves..................................... 44
3.3.4 HelicalCurves ...................................... 44
3.3.5 Congruence......................................... 45
3.3.6 TwoMoreExamples ................................. 47
ProblemSetC.Curves ........................................... 51
GlossaryofMATLABCommands ................................. 59
4 Kinematics .................................................... 61
4.1 Newton’sLawsofMotion ................................... 61
4.2 Kepler’sLawsofPlanetaryMotion............................ 64
4.3 StudyingEquationsofMotionwithMATLAB .................. 65
ProblemSetD.Kinematics ....................................... 67
GlossaryofMATLABCommands ................................. 72
Reference...................................................... 73
5 DirectionalDerivatives.......................................... 75
5.1 VisualizingFunctionsofTwoVariables ........................ 75
5.1.1 Three-DimensionalGraphs ............................ 76
5.1.2 GraphingLevelCurves ............................... 77
5.2 TheGradientofaFunctionofTwoVariables.................... 80
5.2.1 PartialDerivativesandtheGradient..................... 80
5.2.2 DirectionalDerivatives ............................... 82
5.3 FunctionsofThreeorMoreVariables.......................... 85
ProblemSetE.DirectionalDerivatives ............................. 89
GlossaryofMATLABCommandsandOptions ...................... 94
OptionstoMATLABCommands .................................. 94
6 GeometryofSurfaces........................................... 95
6.1 TheConceptofaSurface .................................... 95
6.1.1 BasicExamples...................................... 96
6.2 TheImplicitFunctionTheorem............................... 102
6.3 GeometricInvariants........................................ 105
6.4 CurvatureCalculationswithMATLAB ........................ 112
ProblemSetF.Surfaces .......................................... 115
GlossaryofMATLABCommandsandOptions ...................... 121
OptionstoMATLABCommands .................................. 121
References..................................................... 121
Contents xi
7 OptimizationinSeveralVariables................................ 123
7.1 TheOne-VariableCase...................................... 123
7.1.1 AnalyticMethods.................................... 123
7.1.2 NumericalMethods .................................. 124
7.1.3 Newton’sMethod .................................... 125
7.2 FunctionsofTwoVariables .................................. 127
7.2.1 SecondDerivativeTest................................ 128
7.2.2 SteepestDescent..................................... 130
7.2.3 MultivariableNewton’sMethod ....................... 133
7.3 ThreeorMoreVariables..................................... 134
7.4 ConstrainedOptimizationandLagrangeMultipliers.............. 136
ProblemSetG.Optimization...................................... 139
GlossaryofMATLABCommands ................................. 146
8 MultipleIntegrals.............................................. 147
8.1 AutomationandIntegration .................................. 147
8.1.1 RegionsinthePlane.................................. 148
8.1.2 ViewingSimpleRegions .............................. 151
8.1.3 PolarRegions ....................................... 152
8.2 AlgorithmsforNumericalIntegration.......................... 156
8.2.1 Algorithms for Numerical Integration in a Single
Variable ............................................ 156
8.2.2 AlgorithmsforNumericalMultipleIntegration ........... 157
8.3 ViewingSolidRegions ...................................... 160
8.4 AMoreComplicatedExample ............................... 165
8.5 CylindricalCoordinates ..................................... 169
8.6 MoreGeneralChangesofCoordinates......................... 170
ProblemSetH.MultipleIntegrals.................................. 173
GlossaryofMATLABCommands ................................. 183
9 MultidimensionalCalculus...................................... 185
9.1 TheFundamentalTheoremofLineIntegrals.................... 186
9.2 Green’sTheorem........................................... 190
9.3 Stokes’Theorem ........................................... 192
9.4 TheDivergenceTheorem .................................... 194
9.5 VectorCalculusandPhysics ................................. 196
ProblemSetI.MultivariableCalculus .............................. 199
GlossaryofMATLABCommands ................................. 202
OptionstoMATLABCommands .................................. 203
References..................................................... 203