Mathematics For
Physical Science
And Engineering
Symbolic Computing Applications In Maple And Mathematica
Frank E. Harris
Instructor’s
Manual
, Mathematics For Physical Science And Engineering:
Symbolic Computing Applications In Maple And Mathematica
Instructor’s Manual
Frank E. Harris
Contents
0 Introduction 1
,1 Computers, Science, And Engineering 3
1.1 Computing: Historical Note ................................................................................................ 3
1.2 Basics Of Symbolic Computing .......................................................................................... 3
1.3 Symbolic Computation Programs .................................................................................... 8
1.4 Procedures ................................................................................................................................. 10
1.5 Graphs And Tables ............................................................................................................... 12
1.6 Summary: Symbolic Computing..................................................................................... 15
2 Infinite Series 16
2.1 Definition Of Series .............................................................................................................. 16
2.2 Tests For Convergence ........................................................................................................ 18
2.3 Alternating Series ................................................................................................................. 20
2.4 Operations On Series ........................................................................................................... 21
2.5 Series Of Functions................................................................................................................ 22
2.6 Binomial Theorem ................................................................................................................ 26
2.7 Some Important Series....................................................................................................... 29
2.8 Some Applications Of Series ........................................................................................... 29
2.9 Bernoulli Numbers................................................................................................................ 30
2.10 Asymptotic Series ................................................................................................................. 32
2.11 Euler-Maclaurin Formula .................................................................................................. 32
3 Complex Numbers And Functions 35
3.1 Introduction .............................................................................................................................. 35
3.2 Functions In The Complex Domain ............................................................................. 36
3.3 The Complex Plane .............................................................................................................. 38
3.4 Circular And Hyperbolic Functions ............................................................................. 40
3.5 Multiple-Valued Functions ............................................................................................... 43
4 Vectors And Matrices 47
4.1 Basics Of Vector Algebra.................................................................................................... 47
4.2 Dot Product............................................................................................................................... 50
4.3 Symbolic Computing, Vectors ........................................................................................ 51
Ii Contents
4.4 Matrices .............................................................................................................................................. 54
4.5 Symbolic Computing, Matrices .............................................................................................. 57
4.6 Systems Of Linear Equations .................................................................................................. 61
4.7 Determinants .................................................................................................................................... 63
4.8 Applications Of Determinants................................................................................................. 64
5 Matrix Transformations 70
5.1 Vectors In Rotated Systems..................................................................................................... 70
5.2 Vectors Under Coordinate Reflections ............................................................................... 72
5.3 Transforming Matrix Equations ............................................................................................ 72
5.4 Gram-Schmidt Orthogonalization ......................................................................................... 73
5.5 Matrix Eigenvalue Problems ................................................................................................... 74
5.6 Hermitian Eigenvalue Problems ............................................................................................ 75
, 5.7 Matrix Diagonalization ............................................................................................................... 75
5.8 Matrix Invariants........................................................................................................................... 77
6 Multidimensional Problems 79
6.1 Partial Differentiation ................................................................................................................. 79
6.2 Extrema And Saddle Points..................................................................................................... 82
6.3 Curvilinear Coordinate Systems ........................................................................................... 83
6.4 Multiple Integrals .......................................................................................................................... 85
6.5 Line And Surface Integrals ....................................................................................................... 88
6.6 Rearrangement Of Double Series .......................................................................................... 90
6.7 Dirac Delta Function ................................................................................................................... 91
7 Vector Analysis 93
7.1 Vector Algebra................................................................................................................................. 93
7.2 Vector Differential Operators .................................................................................................. 99
7.3 Vector Differential Operators: Further Properties ................................................... 103
7.4 Integral Theorems ...................................................................................................................... 106
7.5 Potential Theory ......................................................................................................................... 108
7.6 Vectors In Curvilinear Coordinates .................................................................................. 111
8 Tensor Analysis 119
8.1 Cartesian Tensors ...................................................................................................................... 119
8.2 Pseudotensors And Dual Tensors...................................................................................... 124
8.3 Noncartesian Tensors .............................................................................................................. 125
8.4 Symbolic Computation ............................................................................................................. 128
9 Gamma Function 130
9.1 Definition And Properties ...................................................................................................... 130
9.2 Digamma And Polygamma Functions ............................................................................. 132
9.3 Stirling’s Formula ....................................................................................................................... 135
9.4 Beta Function ............................................................................................................................... 136
9.5 Error Function.............................................................................................................................. 140
9.6 Exponential Integral ................................................................................................................. 142
Physical Science
And Engineering
Symbolic Computing Applications In Maple And Mathematica
Frank E. Harris
Instructor’s
Manual
, Mathematics For Physical Science And Engineering:
Symbolic Computing Applications In Maple And Mathematica
Instructor’s Manual
Frank E. Harris
Contents
0 Introduction 1
,1 Computers, Science, And Engineering 3
1.1 Computing: Historical Note ................................................................................................ 3
1.2 Basics Of Symbolic Computing .......................................................................................... 3
1.3 Symbolic Computation Programs .................................................................................... 8
1.4 Procedures ................................................................................................................................. 10
1.5 Graphs And Tables ............................................................................................................... 12
1.6 Summary: Symbolic Computing..................................................................................... 15
2 Infinite Series 16
2.1 Definition Of Series .............................................................................................................. 16
2.2 Tests For Convergence ........................................................................................................ 18
2.3 Alternating Series ................................................................................................................. 20
2.4 Operations On Series ........................................................................................................... 21
2.5 Series Of Functions................................................................................................................ 22
2.6 Binomial Theorem ................................................................................................................ 26
2.7 Some Important Series....................................................................................................... 29
2.8 Some Applications Of Series ........................................................................................... 29
2.9 Bernoulli Numbers................................................................................................................ 30
2.10 Asymptotic Series ................................................................................................................. 32
2.11 Euler-Maclaurin Formula .................................................................................................. 32
3 Complex Numbers And Functions 35
3.1 Introduction .............................................................................................................................. 35
3.2 Functions In The Complex Domain ............................................................................. 36
3.3 The Complex Plane .............................................................................................................. 38
3.4 Circular And Hyperbolic Functions ............................................................................. 40
3.5 Multiple-Valued Functions ............................................................................................... 43
4 Vectors And Matrices 47
4.1 Basics Of Vector Algebra.................................................................................................... 47
4.2 Dot Product............................................................................................................................... 50
4.3 Symbolic Computing, Vectors ........................................................................................ 51
Ii Contents
4.4 Matrices .............................................................................................................................................. 54
4.5 Symbolic Computing, Matrices .............................................................................................. 57
4.6 Systems Of Linear Equations .................................................................................................. 61
4.7 Determinants .................................................................................................................................... 63
4.8 Applications Of Determinants................................................................................................. 64
5 Matrix Transformations 70
5.1 Vectors In Rotated Systems..................................................................................................... 70
5.2 Vectors Under Coordinate Reflections ............................................................................... 72
5.3 Transforming Matrix Equations ............................................................................................ 72
5.4 Gram-Schmidt Orthogonalization ......................................................................................... 73
5.5 Matrix Eigenvalue Problems ................................................................................................... 74
5.6 Hermitian Eigenvalue Problems ............................................................................................ 75
, 5.7 Matrix Diagonalization ............................................................................................................... 75
5.8 Matrix Invariants........................................................................................................................... 77
6 Multidimensional Problems 79
6.1 Partial Differentiation ................................................................................................................. 79
6.2 Extrema And Saddle Points..................................................................................................... 82
6.3 Curvilinear Coordinate Systems ........................................................................................... 83
6.4 Multiple Integrals .......................................................................................................................... 85
6.5 Line And Surface Integrals ....................................................................................................... 88
6.6 Rearrangement Of Double Series .......................................................................................... 90
6.7 Dirac Delta Function ................................................................................................................... 91
7 Vector Analysis 93
7.1 Vector Algebra................................................................................................................................. 93
7.2 Vector Differential Operators .................................................................................................. 99
7.3 Vector Differential Operators: Further Properties ................................................... 103
7.4 Integral Theorems ...................................................................................................................... 106
7.5 Potential Theory ......................................................................................................................... 108
7.6 Vectors In Curvilinear Coordinates .................................................................................. 111
8 Tensor Analysis 119
8.1 Cartesian Tensors ...................................................................................................................... 119
8.2 Pseudotensors And Dual Tensors...................................................................................... 124
8.3 Noncartesian Tensors .............................................................................................................. 125
8.4 Symbolic Computation ............................................................................................................. 128
9 Gamma Function 130
9.1 Definition And Properties ...................................................................................................... 130
9.2 Digamma And Polygamma Functions ............................................................................. 132
9.3 Stirling’s Formula ....................................................................................................................... 135
9.4 Beta Function ............................................................................................................................... 136
9.5 Error Function.............................................................................................................................. 140
9.6 Exponential Integral ................................................................................................................. 142
