MATHEMATICS FORP B B
HYSICAL SCIENCE A B B
ND ENGINEERING
B
Symbolic Computing Applications in Maple and Mathematica
B B B B B B
Frank E. Harris
B B
INSTRUCTOR’SM
ANUAL
B
, Mathematics for Physical Science and Engineering: Symb
B B B B B B
olic Computing Applications in Maple and Mathematica
B B B B B B
Instructor’s Manual B
FrankB E.B Harris
UniversityBofBUtah,BSaltBLakeBBCity,BBUTB
andB UniversityB ofB Florida,B Gainesville,B F
L
,AcademicB PressB isB anB imprintB ofB Elsevier
225B WymanB Street,B Waltham,B MAB 02451,B USA
TheB Boulevard,B LangfordB Lane,B Kidlington,B Oxford,B OX5B 1GB,B UKBCopyri
ghtB ©B2014B ElsevierB Inc.B AllB rightsB reserved
NoB partB ofB thisB publicationB mayB beB reproduced,B storedB inB aB retrievalB systemB orB transmi
ttedBinBanyBformBorBbyBanyBmeansBelectronic,Bmechanical,Bphotocopying,BrecordingBorBotherw
iseBwithoutB theB priorB writtenB permissionB ofB theB publisher.
PermissionsB mayB beB soughtB directlyB fromB Elsevier’sB ScienceB &B TechnologyB RightsB Departm
entBinBOxford,BUK:Bphone+B(B44)B(0)B1865B843830;Bfax +B(B44)B(0)B1865B853333;Bemail:Bpermissi
yourB requestB onlineB byB visitingB theB Else
vierBwebBsiteBatBhttp://elsevier.com/locate/permissions,BandBselectingBObtainingBpermissionB
toBuseBElsevierB material.
Notice
NoBresponsibilityBisBassumedBbyBtheBpublisherBforBanyBinjuryBand/orBdamageBtoBpersonsBorBp
ropertyBasBaBmatterBofBproductsBliability,BnegligenceBorBotherwise,BorBfromBanyBuseBorBopera-
BtionB ofB anyB methods,B products,B instructionsB orB ideasB containedB inB theB materialB herein.
BritishBLibraryBCataloguingBinBPublicationBData
AB catalogueB recordB forB thisB bookB isB availableB fromB theB BritishB Library
LibraryBofBCongressBCataloging-in-PublicationBData
AB catalogB recordB forB thisB bookB isB availableB fromB theB LibraryB ofB Congress
ISBN:B978-0-12-801000-6
For information on all Academic Press publications
visit our web site at store.elsevier.com
PrintedB andB boundB inB USA
14B 15B 16B 17B 18BBBB 10B 9B 8B 7B 6B 5B 4B 3B 2B 1
,Contents
0 Introduction 1
1 Computers,BScience,BandBEngineering 3
1.1 Computing:B HistoricalB Note ................................................................................................3
1.2 BasicsB ofB SymbolicB Computing.........................................................................................3
1.3 SymbolicB ComputationB Programs ...................................................................................8
1.4 Procedures................................................................................................................................. 10
1.5 GraphsB andB Tables ............................................................................................................... 12
1.6 Summary:B SymbolicB Computing.................................................................................... 15
2 InfiniteBSeries 16
2.1 DefinitionB ofB Series.............................................................................................................. 16
2.2 TestsB forB Convergence ....................................................................................................... 18
2.3 AlternatingB Series ................................................................................................................. 20
2.4 OperationsB onB Series .......................................................................................................... 21
2.5 SeriesB ofB Functions .............................................................................................................. 22
2.6 BinomialB Theorem................................................................................................................ 26
2.7 SomeB ImportantB Series ..................................................................................................... 29
2.8 SomeB ApplicationsB ofB Series .......................................................................................... 29
2.9 BernoulliB Numbers............................................................................................................... 30
2.10 AsymptoticB Series ................................................................................................................. 32
2.11 Euler-MaclaurinB Formula ................................................................................................. 32
3 ComplexBNumbersBandBFunctions 35
3.1 Introduction ............................................................................................................................. 35
3.2 FunctionsB inB theB ComplexB Domain ............................................................................ 36
3.3 TheB ComplexB Plane .................................................................................................... 38
3.4 CircularB andB HyperbolicB Functions ............................................................................ 40
3.5 Multiple-ValuedB Functions ............................................................................................... 43
4 VectorsBandBMatrices 47
4.1 BasicsB ofB VectorB Algebra .................................................................................................. 47
4.2 DotB Product .............................................................................................................................. 50
4.3 SymbolicB Computing,B Vectors ............................................................................... 51
, ii CONTENTS
4.4 Matrices ............................................................................................................................................. 54
4.5 SymbolicB Computing,B Matrices ............................................................................................ 57
4.6 SystemsB ofB LinearB Equations................................................................................................ 61
4.7 Determinants .................................................................................................................................. 63
4.8 ApplicationsB ofB Determinants............................................................................................... 64
5 MatrixBTransformations 70
5.1 VectorsB inB RotatedB Systems .................................................................................................. 70
5.2 VectorsB underB CoordinateB Reflections ............................................................................ 72
5.3 TransformingB MatrixB Equations .......................................................................................... 72
5.4 Gram-SchmidtB Orthogonalization........................................................................................ 73
5.5 MatrixB EigenvalueB Problems ................................................................................................. 74
5.6 HermitianB EigenvalueB Problems ......................................................................................... 75
5.7 MatrixB Diagonalization.............................................................................................................. 75
5.8 MatrixB Invariants ......................................................................................................................... 77
6 MultidimensionalBProblems 79
6.1 PartialB Differentiation ............................................................................................................... 79
6.2 ExtremaB andB SaddleB Points ................................................................................................... 82
6.3 CurvilinearB CoordinateB Systems.......................................................................................... 83
6.4 MultipleB Integrals ........................................................................................................................ 85
6.5 LineB andB SurfaceB Integrals..................................................................................................... 88
6.6 RearrangementB ofB DoubleB Series ....................................................................................... 90
6.7 DiracB DeltaB Function ................................................................................................................. 91
7 VectorBAnalysis 93
7.1 VectorB Algebra............................................................................................................................... 93
7.2 VectorB DifferentialB Operators ............................................................................................... 99
7.3 VectorB DifferentialB Operators:B FurtherB Properties ................................................103
7.4 IntegralB Theorems .....................................................................................................................106
7.5 PotentialB Theory.........................................................................................................................108
7.6 VectorsB inB CurvilinearB Coordinates ................................................................................111
8 TensorBAnalysis 119
8.1 CartesianB Tensors ......................................................................................................................119
8.2 PseudotensorsB andB DualB Tensors ....................................................................................124
8.3 NonCartesianB Tensors .............................................................................................................125
8.4 SymbolicB Computation ............................................................................................................128
9 GammaBFunction 130
9.1 DefinitionB andB Properties .....................................................................................................130
9.2 DigammaB andB PolygammaB Functions ............................................................................132
9.3 Stirling’sB Formula ......................................................................................................................135
9.4 BetaB Function...............................................................................................................................136
9.5 ErrorB Function.............................................................................................................................140
9.6 ExponentialB Integral.................................................................................................................142
HYSICAL SCIENCE A B B
ND ENGINEERING
B
Symbolic Computing Applications in Maple and Mathematica
B B B B B B
Frank E. Harris
B B
INSTRUCTOR’SM
ANUAL
B
, Mathematics for Physical Science and Engineering: Symb
B B B B B B
olic Computing Applications in Maple and Mathematica
B B B B B B
Instructor’s Manual B
FrankB E.B Harris
UniversityBofBUtah,BSaltBLakeBBCity,BBUTB
andB UniversityB ofB Florida,B Gainesville,B F
L
,AcademicB PressB isB anB imprintB ofB Elsevier
225B WymanB Street,B Waltham,B MAB 02451,B USA
TheB Boulevard,B LangfordB Lane,B Kidlington,B Oxford,B OX5B 1GB,B UKBCopyri
ghtB ©B2014B ElsevierB Inc.B AllB rightsB reserved
NoB partB ofB thisB publicationB mayB beB reproduced,B storedB inB aB retrievalB systemB orB transmi
ttedBinBanyBformBorBbyBanyBmeansBelectronic,Bmechanical,Bphotocopying,BrecordingBorBotherw
iseBwithoutB theB priorB writtenB permissionB ofB theB publisher.
PermissionsB mayB beB soughtB directlyB fromB Elsevier’sB ScienceB &B TechnologyB RightsB Departm
entBinBOxford,BUK:Bphone+B(B44)B(0)B1865B843830;Bfax +B(B44)B(0)B1865B853333;Bemail:Bpermissi
yourB requestB onlineB byB visitingB theB Else
vierBwebBsiteBatBhttp://elsevier.com/locate/permissions,BandBselectingBObtainingBpermissionB
toBuseBElsevierB material.
Notice
NoBresponsibilityBisBassumedBbyBtheBpublisherBforBanyBinjuryBand/orBdamageBtoBpersonsBorBp
ropertyBasBaBmatterBofBproductsBliability,BnegligenceBorBotherwise,BorBfromBanyBuseBorBopera-
BtionB ofB anyB methods,B products,B instructionsB orB ideasB containedB inB theB materialB herein.
BritishBLibraryBCataloguingBinBPublicationBData
AB catalogueB recordB forB thisB bookB isB availableB fromB theB BritishB Library
LibraryBofBCongressBCataloging-in-PublicationBData
AB catalogB recordB forB thisB bookB isB availableB fromB theB LibraryB ofB Congress
ISBN:B978-0-12-801000-6
For information on all Academic Press publications
visit our web site at store.elsevier.com
PrintedB andB boundB inB USA
14B 15B 16B 17B 18BBBB 10B 9B 8B 7B 6B 5B 4B 3B 2B 1
,Contents
0 Introduction 1
1 Computers,BScience,BandBEngineering 3
1.1 Computing:B HistoricalB Note ................................................................................................3
1.2 BasicsB ofB SymbolicB Computing.........................................................................................3
1.3 SymbolicB ComputationB Programs ...................................................................................8
1.4 Procedures................................................................................................................................. 10
1.5 GraphsB andB Tables ............................................................................................................... 12
1.6 Summary:B SymbolicB Computing.................................................................................... 15
2 InfiniteBSeries 16
2.1 DefinitionB ofB Series.............................................................................................................. 16
2.2 TestsB forB Convergence ....................................................................................................... 18
2.3 AlternatingB Series ................................................................................................................. 20
2.4 OperationsB onB Series .......................................................................................................... 21
2.5 SeriesB ofB Functions .............................................................................................................. 22
2.6 BinomialB Theorem................................................................................................................ 26
2.7 SomeB ImportantB Series ..................................................................................................... 29
2.8 SomeB ApplicationsB ofB Series .......................................................................................... 29
2.9 BernoulliB Numbers............................................................................................................... 30
2.10 AsymptoticB Series ................................................................................................................. 32
2.11 Euler-MaclaurinB Formula ................................................................................................. 32
3 ComplexBNumbersBandBFunctions 35
3.1 Introduction ............................................................................................................................. 35
3.2 FunctionsB inB theB ComplexB Domain ............................................................................ 36
3.3 TheB ComplexB Plane .................................................................................................... 38
3.4 CircularB andB HyperbolicB Functions ............................................................................ 40
3.5 Multiple-ValuedB Functions ............................................................................................... 43
4 VectorsBandBMatrices 47
4.1 BasicsB ofB VectorB Algebra .................................................................................................. 47
4.2 DotB Product .............................................................................................................................. 50
4.3 SymbolicB Computing,B Vectors ............................................................................... 51
, ii CONTENTS
4.4 Matrices ............................................................................................................................................. 54
4.5 SymbolicB Computing,B Matrices ............................................................................................ 57
4.6 SystemsB ofB LinearB Equations................................................................................................ 61
4.7 Determinants .................................................................................................................................. 63
4.8 ApplicationsB ofB Determinants............................................................................................... 64
5 MatrixBTransformations 70
5.1 VectorsB inB RotatedB Systems .................................................................................................. 70
5.2 VectorsB underB CoordinateB Reflections ............................................................................ 72
5.3 TransformingB MatrixB Equations .......................................................................................... 72
5.4 Gram-SchmidtB Orthogonalization........................................................................................ 73
5.5 MatrixB EigenvalueB Problems ................................................................................................. 74
5.6 HermitianB EigenvalueB Problems ......................................................................................... 75
5.7 MatrixB Diagonalization.............................................................................................................. 75
5.8 MatrixB Invariants ......................................................................................................................... 77
6 MultidimensionalBProblems 79
6.1 PartialB Differentiation ............................................................................................................... 79
6.2 ExtremaB andB SaddleB Points ................................................................................................... 82
6.3 CurvilinearB CoordinateB Systems.......................................................................................... 83
6.4 MultipleB Integrals ........................................................................................................................ 85
6.5 LineB andB SurfaceB Integrals..................................................................................................... 88
6.6 RearrangementB ofB DoubleB Series ....................................................................................... 90
6.7 DiracB DeltaB Function ................................................................................................................. 91
7 VectorBAnalysis 93
7.1 VectorB Algebra............................................................................................................................... 93
7.2 VectorB DifferentialB Operators ............................................................................................... 99
7.3 VectorB DifferentialB Operators:B FurtherB Properties ................................................103
7.4 IntegralB Theorems .....................................................................................................................106
7.5 PotentialB Theory.........................................................................................................................108
7.6 VectorsB inB CurvilinearB Coordinates ................................................................................111
8 TensorBAnalysis 119
8.1 CartesianB Tensors ......................................................................................................................119
8.2 PseudotensorsB andB DualB Tensors ....................................................................................124
8.3 NonCartesianB Tensors .............................................................................................................125
8.4 SymbolicB Computation ............................................................................................................128
9 GammaBFunction 130
9.1 DefinitionB andB Properties .....................................................................................................130
9.2 DigammaB andB PolygammaB Functions ............................................................................132
9.3 Stirling’sB Formula ......................................................................................................................135
9.4 BetaB Function...............................................................................................................................136
9.5 ErrorB Function.............................................................................................................................140
9.6 ExponentialB Integral.................................................................................................................142
