Vector analysis: An introduction to vector-methods and their various applications

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Preface to the Second Edition......Page 12
CONTENTS......Page 14
2. Graphical Representation of a Vector......Page 24
3. Equality of Vectors - Negative Vector - Unit Vector - Reciprocal Vector......Page 25
4. Composition of Vectors - Addition and Subtraction - Vector Sum as an Integration......Page 27
5. Scalar and Vector Fields - Point-Function - Definition of Lame - Continuity of Scalar and Vector Functions......Page 29
6. Decomposition of Vectors......Page 31
7. The Unit Vectors I j k......Page 32
8. Vector Equations - Equations of Straight Line and Plane.......Page 34
9. Condition that Three Vectors Terminate in a Straight Line - Examples......Page 36
11. Plane Passing through Ends of Three Given Vectors......Page 39
13. To Divide a Line in a Given Ratio - Centroid......Page 41
14. Relations Independent of the Origin - General Condition......Page 44
EXERCISES AND PROBLEMS......Page 45
15. Scalar or Dot Product - Laws of the Scalar Product......Page 51
16. Line-Integral of a Vector......Page 54
17. Surface-Integral of a Vector......Page 55
18. Vector or Cross Product - Definition......Page 57
19 Distributive Law of Vector Products - Physical Proof......Page 58
20. Cartesian Expansion of the Vector Product......Page 61
21. Applications to Mechanics - Moment......Page 62
23. Composition of Angular Velocities......Page 64
EXERCISES AND PROBLEMS......Page 66
24. Possible Combinations of Three Vectors......Page 71
26. Condition that Three Vectors lie in a Plane - Manipulation of Scalar Magnitudes of Vectors......Page 73
27. Triple Vector Product q - ax(bxc) - Expansion and Proof.......Page 74
28. Demonstration by Cartesian Expansion......Page 76
29. Third Proof......Page 77
30. Products of More than Three Vectors......Page 78
31. Reciprocal System of Vectors......Page 80
32. Plane Normal to a and Passing through End of b - Plane through Ends of Three Given Vectors - Vector Perpen-dicular from Origin to a Plane......Page 81
33. Line through End of b Parallel to a......Page 83
34. Circle and Sphere......Page 84
34a. Resolution of System of Forces Acting on a Rigid Body Central Axis - Minimum Couple......Page 86
EXERCISES AND PROBLEMS......Page 89
35. Two Ways in which a Vector may Vary - Differentiation with Respect to Scalar Variables......Page 93
36. Differentiation of Scalar and Vector Products......Page 95
37. Applications to Geometry - Tangent and Normal......Page 96
38. Curvature - Osculating Plane - Tortuosity-Geodetic Lines on a Surface......Page 99
39. Equations of Surfaces - Curvilinear Coordinates - Orthogonal System......Page 102
40. Applications to Kinematics of a Particle - Hodographs -Equations of Hodographs......Page 103
41. Integration with Respect to a Scalar Variable - Orbit of a Planet - Harmonic Motion - Ellipse......Page 106
42. Hodograph and Orbit under Newtonian Forces......Page 110
43. Partial Differentiation - Origin of the Operator V......Page 113
EXERCISES AND PROBLEMS......Page 114
44. Scalar and Vector Fields......Page 117
45. Scalar and Vector Functions of Position - Mathematical and Physical Discontinuities......Page 118
46. Potential - Level or Equipotential Surfaces - Relation  between Force and Potential......Page 121
47. V applied to a Scalar Function - Gradient - Independence  of Axes - Fourier's Law......Page 125
48. V applied to Scalar Functions - Effect of V on Scalar Product......Page 127
49. The Operator S ,  V, or Directional Derivative - Total Derivative......Page 129
50. Directional Derivative of a Vector - V applied to a Vector Point-Function......Page 130
51. Divergence - The Operator V......Page 132
52. The Divergence Theorem - Examples - Equation of Flow of Heat......Page 135
53. Equation of Continuity - Solenoidal Distribution of a Vector......Page 139
54. Curl- The Operator Vx - Example of Curl......Page 140
55. Motion of Rotation without Curl - Irrotational Motion......Page 142
56. V, Vx applied to Various Functions - Proofs of Formulae......Page 143
58. Stokes' Theorem......Page 147
59. Condition for Vanishing of the Curl - Conservative System of Forces......Page 150
60. Condition for a Perfect Differential......Page 152
62. Euler's Theorem on Homogeneous Functions......Page 154
63. Operators Involving V Twice - Possible Combinations - The Operator V2 - V -V......Page 156
64. Differentiation of rm by V......Page 158
EXERCISES AND PROBLEMS......Page 159
65. Gauss's Theorem - Solid Angle - Gauss's Theorem for the  Plane - Second Proof......Page 161
66. The Potential Function - Poisson's and Laplace's Equations  - Harmonic Function......Page 166
68. Green's Formulae - Green's Function......Page 171
69. Solution of Poisson's Equation - The Integrating Operator ( Pot = f ( )dv......Page 175
70.  Vector-Potential......Page 176
71. Separation of a Vector-Function into Solenoidal and Lamellar  Components - Other Systems of Units......Page 177
72. Energy in Terms of Potential......Page 179
73. Energy in Terms of Field Intensity......Page 180
74. Surface and Volume Density in Terms of Polarization......Page 182
75. Electro-Magnetic Field - Maxwell's Equations......Page 183
76. Equation of Propagation of Electro-Magnetic Waves......Page 186
77. Poynting's Theorem - Radiant Vector    :......Page 187
78. Magnetic Field due to a Current......Page 188
79. Mechanical Force on an Element of Current......Page 190
80. Theorem on Line Integral of the Normal Component of a Vector Function......Page 191
81. Electric Field at any Point due to a Current......Page 193
82. Mutual Energy of Circuits - Inductance - Neumann's  Integral......Page 194
83. Vector-Potential of a Current - Mutual Energy of Systems of Conductors - Integration Theorem......Page 196
84. Mutual and Self-Energies of Two Circuits......Page 198
EXERCISES AND PROBLEMS......Page 199
85. Equations of Motion of a Rigid Body-D'Alembert's Equation - Equations of Translation - Motion of Center of  Mass......Page 201
86. Equations of Rotation - Kinetic Energy of Rotation  Moment of Inertia......Page 203
87. Linear Vector-Function -Instantaneous Axis......Page 205
88. Motion of Rotation under No Forces-Poinsot Ellipsoid - Moments and Products of Inertia - Coordinates of a Linear Vector-Function -Principal Moments of Inertia Principal Axes......Page 207
89. Geometrical Representation of the Motion - Invariable Plane - Invariable Axis......Page 214
90. Polhode and Herpolhode Curves - Permanent Axes -Equations of Polhode and Herpolhode......Page 215
91. Moving Axes and Relative Motion - Theorem of Coriolis.......Page 217
92. Transformation of Equations of Motion-Centrifugal Couple - Gyroscope......Page 221
93. Euler's Equations of Motion......Page 222
94. Analytical Solution of Euler's Equations under No Impressed Forces......Page 223
95. Hamilton's Principle - Lagrangian Function......Page 225
96. Extension of Vector to More than Three Dimensions - Definitions......Page 227
97. Lagrange's Generalized Equations of Motion - The Operator VL = 0 Contains the Whole of Mechanics......Page 228
98. Hydrodynamics - Fundamental Equations - Equation of  Continuity - Euler's Equations of Motion of a Fluid......Page 230
99. Transformations of the Equations of Motion......Page 234
101. Vortex Motion - Non-creatable in a Frictionless System Helmholtz's Equations......Page 235
102. Circulation - Definition......Page 237
103. Velocity-Potential - Circulation Invariable in a Frictionless Fluid......Page 239
EXERCISES AND PROBLEMS......Page 240
Grassmann......Page 244
Comparison of Formulae in Different Notations......Page 245
Notation of this Book......Page 247
Vectors......Page 252
Vector and Scalar Products - Products of Two Vectors......Page 253
Products of Three Vectors......Page 254
The Operator V, del......Page 256
Linear Vector Function......Page 260
Note on Different Varieties of Vectors......Page 263
normal and Rectifying Plane for a Space Curve......Page 265
Frenet's Formula; for a Space Curve......Page 267
Motion of an Electron in a Uniform Magnetic Field......Page 268
Two Proofs of Stokes' Theorem......Page 272
Proof of Gauss's Theorem......Page 274
Other Integration Theorems......Page 275
Index......Page 278