Mathematica by Example (3rd Ed.) - Elsevier Academic Press

585 Pages·2016·20.22 MB·English

by  Abell

Mathematica By Example
Third Edition
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Mathematica By Example
Third Edition
Martha L. Abell and James P. Braselton
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03 04 05 06 07 08 9 8 7 6 5 4 3 2 1
Contents
Preface ix
1 GettingStarted 1
1.1 IntroductiontoMathematica . . . . . . . . . . . . . . . . . . . . . . 1
ANoteRegardingDifferentVersionsofMathematica . . . . . . . . 3
1.1.1 GettingStartedwithMathematica . . . . . . . . . . . . . . . 3
Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 LoadingPackages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
AWordofCaution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 GettingHelpfromMathematica. . . . . . . . . . . . . . . . . . . . . 14
MathematicaHelp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
TheMathematicaMenu . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 BasicOperationsonNumbers,Expressions,andFunctions 23
2.1 NumericalCalculationsandBuilt-InFunctions . . . . . . . . . . . . 23
2.1.1 NumericalCalculations . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Built-InConstants . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.3 Built-InFunctions . . . . . . . . . . . . . . . . . . . . . . . . 27
AWordofCaution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 ExpressionsandFunctions:ElementaryAlgebra . . . . . . . . . . . 31
2.2.1 BasicAlgebraicOperationsonExpressions . . . . . . . . . . 31
2.2.2 NamingandEvaluatingExpressions . . . . . . . . . . . . . 37
TwoWordsofCaution . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 DefiningandEvaluatingFunctions . . . . . . . . . . . . . . 39
2.3 GraphingFunctions,Expressions,andEquations . . . . . . . . . . . 45
v
vi Contents
2.3.1 FunctionsofaSingleVariable . . . . . . . . . . . . . . . . . 45
2.3.2 ParametricandPolarPlotsinTwoDimensions . . . . . . . 58
2.3.3 Three-DimensionalandContourPlots;GraphingEquations 65
2.3.4 ParametricCurvesandSurfacesinSpace . . . . . . . . . . . 75
2.4 SolvingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4.1 ExactSolutionsofEquations . . . . . . . . . . . . . . . . . . 81
2.4.2 ApproximateSolutionsofEquations . . . . . . . . . . . . . 90
3 Calculus 97
3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.1 UsingGraphsandTablestoPredictLimits . . . . . . . . . . 98
3.1.2 ComputingLimits . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1.3 One-SidedLimits. . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2 DifferentialCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.2.1 DefinitionoftheDerivative . . . . . . . . . . . . . . . . . . . 104
3.2.2 CalculatingDerivatives . . . . . . . . . . . . . . . . . . . . . 107
3.2.3 ImplicitDifferentiation . . . . . . . . . . . . . . . . . . . . . 110
3.2.4 TangentLines . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2.5 TheFirstDerivativeTestandSecondDerivativeTest . . . . 123
3.2.6 AppliedMax/MinProblems . . . . . . . . . . . . . . . . . . 128
3.2.7 Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 138
3.3 IntegralCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.3.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.3.2 TheDefiniteIntegral . . . . . . . . . . . . . . . . . . . . . . . 147
3.3.3 ApproximatingDefiniteIntegrals . . . . . . . . . . . . . . . 153
3.3.4 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.3.5 ArcLength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.3.6 SolidsofRevolution . . . . . . . . . . . . . . . . . . . . . . . 167
3.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.4.1 IntroductiontoSequencesandSeries . . . . . . . . . . . . . 173
3.4.2 ConvergenceTests . . . . . . . . . . . . . . . . . . . . . . . . 178
3.4.3 AlternatingSeries . . . . . . . . . . . . . . . . . . . . . . . . 182
3.4.4 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.4.5 TaylorandMaclaurinSeries . . . . . . . . . . . . . . . . . . 187
3.4.6 Taylor’sTheorem. . . . . . . . . . . . . . . . . . . . . . . . . 192
3.4.7 OtherSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.5 Multi-VariableCalculus . . . . . . . . . . . . . . . . . . . . . . . . . 198
3.5.1 LimitsofFunctionsofTwoVariables . . . . . . . . . . . . . 198
3.5.2 PartialandDirectionalDerivatives . . . . . . . . . . . . . . 201
3.5.3 IteratedIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . 218
Contents vii
4 IntroductiontoListsandTables 229
4.1 ListsandListOperations . . . . . . . . . . . . . . . . . . . . . . . . . 229
4.1.1 DefiningLists . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
4.1.2 PlottingListsofPoints. . . . . . . . . . . . . . . . . . . . . . 233
4.2 ManipulatingLists:MoreonPartandMap . . . . . . . . . . . . . . 248
4.2.1 MoreonGraphingLists;GraphingListsofPointsUsing
GraphicsPrimitives . . . . . . . . . . . . . . . . . . . . . . . 258
4.2.2 MiscellaneousListOperations . . . . . . . . . . . . . . . . . 267
4.3 MathematicsofFinance . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.3.1 CompoundInterest . . . . . . . . . . . . . . . . . . . . . . . 268
4.3.2 FutureValue . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
4.3.3 AnnuityDue . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.3.4 PresentValue . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
4.3.5 DeferredAnnuities. . . . . . . . . . . . . . . . . . . . . . . . 274
4.3.6 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.3.7 MoreonFinancialPlanning. . . . . . . . . . . . . . . . . . . 280
4.4 OtherApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
4.4.1 ApproximatingListswithFunctions . . . . . . . . . . . . . 287
4.4.2 IntroductiontoFourierSeries . . . . . . . . . . . . . . . . . 294
4.4.3 TheMandelbrotSetandJuliaSets . . . . . . . . . . . . . . . 308
5 MatricesandVectors 327
5.1 NestedLists:IntroductiontoMatrices,Vectors,and
MatrixOperations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
5.1.1 DefiningNestedLists,Matrices,andVectors . . . . . . . . . 327
5.1.2 ExtractingElementsofMatrices . . . . . . . . . . . . . . . . 334
5.1.3 BasicComputationswithMatrices . . . . . . . . . . . . . . . 337
5.1.4 BasicComputationswithVectors . . . . . . . . . . . . . . . 342
5.2 LinearSystemsofEquations . . . . . . . . . . . . . . . . . . . . . . . 349
5.2.1 CalculatingSolutionsofLinearSystems
ofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
5.2.2 Gauss–JordanElimination . . . . . . . . . . . . . . . . . . . 355
5.3 SelectedTopicsfromLinearAlgebra . . . . . . . . . . . . . . . . . . 362
5.3.1 FundamentalSubspacesAssociatedwithMatrices . . . . . 362
5.3.2 TheGram–SchmidtProcess . . . . . . . . . . . . . . . . . . . 364
5.3.3 LinearTransformations . . . . . . . . . . . . . . . . . . . . . 370
5.3.4 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . 373
5.3.5 JordanCanonicalForm . . . . . . . . . . . . . . . . . . . . . 377
5.3.6 TheQRMethod . . . . . . . . . . . . . . . . . . . . . . . . . 381
5.4 MaximaandMinimaUsingLinearProgramming . . . . . . . . . . 384
viii Contents
5.4.1 TheStandardFormofaLinearProgramming
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
5.4.2 TheDualProblem . . . . . . . . . . . . . . . . . . . . . . . . 386
5.5 SelectedTopicsfromVectorCalculus . . . . . . . . . . . . . . . . . . 393
5.5.1 Vector-ValuedFunctions . . . . . . . . . . . . . . . . . . . . 393
5.5.2 LineIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
5.5.3 SurfaceIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . 407
5.5.4 ANoteonNonorientability . . . . . . . . . . . . . . . . . . . 411
6 DifferentialEquations 429
6.1 First-OrderDifferentialEquations . . . . . . . . . . . . . . . . . . . 429
6.1.1 SeparableEquations . . . . . . . . . . . . . . . . . . . . . . . 429
6.1.2 LinearEquations . . . . . . . . . . . . . . . . . . . . . . . . . 434
6.1.3 NonlinearEquations . . . . . . . . . . . . . . . . . . . . . . . 444
6.1.4 NumericalMethods . . . . . . . . . . . . . . . . . . . . . . . 448
6.2 Second-OrderLinearEquations . . . . . . . . . . . . . . . . . . . . . 454
6.2.1 BasicTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
6.2.2 ConstantCoefficients . . . . . . . . . . . . . . . . . . . . . . 455
6.2.3 UndeterminedCoefficients . . . . . . . . . . . . . . . . . . . 462
6.2.4 VariationofParameters . . . . . . . . . . . . . . . . . . . . . 467
6.3 Higher-OrderLinearEquations . . . . . . . . . . . . . . . . . . . . . 470
6.3.1 BasicTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
6.3.2 ConstantCoefficients . . . . . . . . . . . . . . . . . . . . . . 470
6.3.3 UndeterminedCoefficients . . . . . . . . . . . . . . . . . . . 473
6.3.4 LaplaceTransformMethods . . . . . . . . . . . . . . . . . . 485
6.3.5 NonlinearHigher-OrderEquations . . . . . . . . . . . . . . 498
6.4 SystemsofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
6.4.1 LinearSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 498
6.4.2 NonhomogeneousLinearSystems . . . . . . . . . . . . . . . 515
6.4.3 NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . . . 519
6.5 SomePartialDifferentialEquations . . . . . . . . . . . . . . . . . . . 538
6.5.1 TheOne-DimensionalWaveEquation . . . . . . . . . . . . . 538
6.5.2 TheTwo-DimensionalWaveEquation. . . . . . . . . . . . . 544
6.5.3 OtherPartialDifferentialEquations . . . . . . . . . . . . . . 556
Preface
Mathematica By Example bridges the gap that exists between the very elementary
handbooks available on Mathematica and those reference books written for the
advancedMathematicausers.MathematicaByExampleisanappropriatereference
forall users ofMathematica and, in particular, forbeginning users like students,
instructors, engineers, business people, and other professionals first learning to
use Mathematica. Mathematica By Example introduces the very basic commands
and includes typical examples of applications of these commands. In addition,
the text also includes commands useful in areas such as calculus, linear algebra,
businessmathematics,ordinaryandpartialdifferentialequations,andgraphics.In
allcases,however,examplesfollowtheintroductionofnewcommands.Readers
from the most elementary to advanced levels will find that the range of topics
coveredaddressestheirneeds.
TakingadvantageofVersion5ofMathematica,MathematicaByExample,Third
Edition,introducesthefundamentalconceptsofMathematicatosolvetypicalprob-
lemsofinteresttostudents,instructors,andscientists.Otherfeaturestohelpmake
MathematicaByExample,ThirdEdition,aseasytouseandasusefulaspossiblein-
cludethefollowing.
1. Version 5 Compatibility. All examples illustrated in Mathematica By
Example,ThirdEdition,werecompletedusingVersion5ofMathematica.
Although most computations can continue to be carried out with earlier
versionsofMathematica,likeVersions2,3,and4,wehavetakenadvan-
tageofthenewfeaturesinVersion5asmuchaspossible.
ix