Vector calculus is the foundation stone on which a vast amount of applied mathematics is based. Topics such as fluid dynamics, solid mechanics and electromagnetism depend heavily on the calculus of vector quantities in three dimensions. This book covers the material in a comprehensive but concise manner, combining mathematical rigour with physical insight. There are many diagrams to illustrate the physical meaning of the mathematical concepts, which is essential for a full understanding of the subject. Each chapter concludes with a summary of the most important points, and there are worked examples that cover all of the material. The final chapter introduces some of the most important applications of vector calculus, including mechanics and electromagnetism.
1. Vector Algebra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Vectors and scalars .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Defini tion of a vector and a scalar . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Addition of vectors ................................ 2
1.1.3 Components of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Dot product .................... . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Applications of the dot product . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Cross product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Applications of the cross product. . . . . . . . . . . . . . . . . . . .. 11
1.4 Scalar triple product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
1.5 Vector triple product ..................................... 16
1.6 Scalar fields and vector fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2. Line, Surface and Volume Integrals. . . . . . . . . . . . . . . . . . . . . . . .. 21
2.1 Applications and methods of integration. . . . . . . . . . . . . . . . . . . .. 21
2.1.1 Examples of the use of integration .................... 21
2.1.2 Integration by substitution .......................... 22
2.1.3 Integration by parts ................................ 23
2.2 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
2.2.1 Introductory example: work done against a force ....... 25
2.2.2 Evaluation of line integrals .......................... 26
2.2.3 Conservative vector fields. . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
2.2.4 Other forms of line integrals . . . . . . . . . . . . . . . . . . . . . . . .. 30
2.3 Surface integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
2.3.1 Introductory example: flow through a pipe. . . . . . . . . . . .. 31
2.3.2 Evaluation of surfa-ce integrals ........................ 33
2.3.3 Other forms of surface integrals . . . . . . . . . . . . . . . . . . . . .. 38
2.4 Volume integrals ......................................... 39
2.4.1 Introductory example: mass of an object with variable density . . . . . . . . . . . . . . . .. 39
2.4.2 Evaluation of volume integrals ............ . . . . . . . . . .. 40
3. Gradient, Divergence and Curl ............... . . . . . . . . . . . . . .. 45
3.1 Partial differentiation and Taylor series. . . . . . . . . . . . . . . . . . . . .. 45
3.1.1 Partial differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
3.1.2 Taylor series in more than one variable. . . . . . . . . . . . . . .. 47
3.2 Gradient of a scalar field ............ . . . . . . . . . . . . . . . . . . .. 48
3.2.1 Gradients, conservative fields and potentials ........... 51
3.2.2 Physical applications of the gradient. . . . . . . . . . . . . . . . .. 52
3.3 Divergence of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
3.3.1 Physical interpretation of divergence. . . . . . . . . . . . . . . . .. 56
3.3.2 Laplacian of a scalar field ................. . . . . . . . .. 56
3.4 Curl of a vector field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58
3.4.1 Physical interpretation of curl. . . . . . . . . . . . . . . . . . . . . . .. 60
3.4.2 Relation between curl and rotation .... . . . . . . . . . . . . . .. 61
3.4.3 Curl and conservative vector fields ................... 61
4. Suffix Notation and its Applications . . . . . . . . . . . . . . . . . . . . . . .. 65
4.1 Introduction to suffix notation ............................. 65
4.2 The Kronecker delta ij ................................... 68
4.3 The alternating tensor fijk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
4.4 Relation between fijk and ij ............................. 72
4.5 Grad, div and curl in suffix notation . . . . . . . . . . . . . . . . . . . . . . .. 74
4.6 Combinations of grad, div and curl ................ . . . . . . . .. 76
4.7 Grad, div and curl applied to products of functions . . . . . . . . .. 78
5. Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
5.1 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
5.1.1 Conservation of mass for a fluid . . . . . . . . . . . . . . . . . . . .. 85
5.1.2 Applications of the divergence theorem. . . . . . . . . . . . . . .. 87
5.1.3 Related theorems linking surface and volume integrals .. 88
5.2 Stokes's theorem ......................................... 91
5.2.1 Applications of Stokes's theorem ..................... 93
5.2.2 Related theorems linking line and surface integrals. . . . .. 95
6. Curvilinear Coordinates 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
6.1 Orthogonal curvilinear coordinates. . . . . . . . . . . . . . . . . . . . . . . . .. 99
6.2 Grad, div and curl in orthogonal curvilinear coordinate systems 104
6.2.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2 Divergence ........................................ 105
6.2.3 Curl.............................................. 106
6.3 Cylindrical polar coordinates. . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . 107
6.4 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7. Cartesian Tensors .......................................... 115
7.1 Coordinate transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Vectors and scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3 Tensors................................................. 119
7.3.1 The quotient rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3.2 Symmetric and anti-symmetric tensors . . . . . . . . . . . . . . . . 122
7.3.3 Isotropic tensors ................................... 123
7.4 Physical examples of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4.1 Ohm's law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4.2 The inertia tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8. Applications of Vector Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.1 Heat transfer ............................................ 132
8.2 Electromagnetism........................................ 134
8.2.1 Electrostatics...................................... 135
8.2.2 Electromagnetic waves in a vacuum. . . . . . . . . . . . . . . . . . . 137
8.3 Continuum mechanics and the stress tensor. . . . . . . . . . . . . . . . . . 140
8.4 Solid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.5 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.5.1 Equation of motion for a fluid . . . . . . . . . . . . . . . . . . . . . . . . 146
8.5.2 The vorticity equation .............................. 147
8.5.3 Bernoulli's equation ................................ 149
Solutions ................................................... 153
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Author(s): Paul C. Matthews
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer
Year: 1998
Language: English
Pages: 182
Preface......Page 6
Table of Contents......Page 8
1.1.1 Defini tion of a vector and a scalar ......Page 12
1.1.2 Addition of vectors ......Page 13
1.1.3 Components of a vector ......Page 14
1.2 Dot product ......Page 15
1.2.1 Applications of the dot product ......Page 18
1.3 Cross product ......Page 20
1.3.1 Applications of the cross product ......Page 22
1.4 Scalar triple product ......Page 25
1.5 Vector triple product ......Page 27
1.6 Scalar fields and vector fields ......Page 28
2.1.1 Examples of the use of integration ......Page 32
2.1.2 Integration by substitution ......Page 33
2.1.3 Integration by parts ......Page 34
2.2.1 Introductory example: work done against a force ......Page 36
2.2.2 Evaluation of line integrals ......Page 37
2.2.3 Conservative vector fields ......Page 39
2.2.4 Other forms of line integrals ......Page 41
2.3.1 Introductory example: flow through a pipe ......Page 42
2.3.2 Evaluation of surfa-ce integrals ......Page 44
2.3.3 Other forms of surface integrals ......Page 49
2.4.1 Introductory example: mass of an object with variable density ......Page 50
2.4.2 Evaluation of volume integrals ......Page 51
3.1.1 Partial differentiation ......Page 56
3.1.2 Taylor series in more than one variable ......Page 58
3.2 Gradient of a scalar field ......Page 59
3.2.1 Gradients, conservative fields and potentials ......Page 62
3.2.2 Physical applications of the gradient ......Page 63
3.3 Divergence of a vector field ......Page 64
3.3.2 Laplacian of a scalar field ......Page 67
3.4 Curl of a vector field ......Page 69
3.4.1 Physical interpretation of curl ......Page 71
3.4.3 Curl and conservative vector fields ......Page 72
4.1 Introduction to suffix notation ......Page 76
4.2 The Kronecker delta ij ......Page 79
4.3 The alternating tensor fijk ......Page 81
4.4 Relation between fijk and ij ......Page 83
4.5 Grad, div and curl in suffix notation ......Page 85
4.6 Combinations of grad, div and curl ......Page 87
4.7 Grad, div and curl applied to products of functions ......Page 89
5.1 Divergence theorem ......Page 94
5.1.1 Conservation of mass for a fluid ......Page 96
5.1.2 Applications of the divergence theorem ......Page 98
5.1.3 Related theorems linking surface and volume integrals ......Page 99
5.2 Stokes's theorem ......Page 102
5.2.1 Applications of Stokes's theorem ......Page 104
5.2.2 Related theorems linking line and surface integrals ......Page 106
6.1 Orthogonal curvilinear coordinates ......Page 110
6.2.1 Gradient ......Page 115
6.2.2 Divergence ......Page 116
6.2.3 Curl ......Page 117
6.3 Cylindrical polar coordinates ......Page 118
6.4 Spherical polar coordinates ......Page 121
7.1 Coordinate transformations ......Page 126
7.2 Vectors and scalars ......Page 128
7.3 Tensors ......Page 130
7.3.1 The quotient rule ......Page 131
7.3.2 Symmetric and anti-symmetric tensors ......Page 133
7.3.3 Isotropic tensors ......Page 134
7.4.1 Ohm's law ......Page 137
7.4.2 The inertia tensor ......Page 138
8. Applications of Vector Calculus ......Page 142
8.1 Heat transfer ......Page 143
8.2 Electromagnetism ......Page 145
8.2.1 Electrostatics ......Page 146
8.2.2 Electromagnetic waves in a vacuum ......Page 148
8.3 Continuum mechanics and the stress tensor ......Page 151
8.4 Solid mechanics ......Page 154
8.5 Fluid mechanics ......Page 156
8.5.1 Equation of motion for a fluid ......Page 157
8.5.2 The vorticity equation ......Page 158
8.5.3 Bernoulli's equation ......Page 160
Solutions ......Page 165
Index ......Page 193
