## C.E. Weatherburn: Advanced Vector Analysis

Published $\text {1924}$, G. Bell and Sons, Ltd..

### Contents

Preface (September $1923$)
Bibliography
Table of Notations
Short Course

CHAPTER $\text {I}$. the differential operators.
1. Vector function of several independent variables
2. Scalar and vector point-functions
3. Gradient of a scalar point-function. The operator $\nabla$
4. Directional derivative. Gradient of a sum or product
5. Gradient of $r^m$ and of $\map F r$
6. Vector point-function. Directional derivative
7. Divergence and curl of a vector. The notation $\nabla \cdot \mathbf F$ and $\nabla \times \mathbf F$
8. Formulæ of expansion
9. Second order differential equation
10. The function $r^m$. Also $\map F r$
11. Orthogonal curvilinear coordinates
12. Curvilinear expressions for grad, div, curl, and $\nabla^2$
Exercises

CHAPTER $\text {II}$. line, surface, and space integrals.
13. Tangential line integral of $\nabla V$
14. The Divergence Theorem of Gauss
15. Theorems deducible from the divergence theorem
16. Stokes's Theorem
17. Certain deductions from the preceding theorems
18. Curvilinear expressions for $\nabla \cdot \mathbf F$ and $\nabla \times \mathbf F$
19. Green's theorem
20. Green's formula. Gauss's integral
21. Green's function for Laplace's equation. Symmetry of this function
Exercises

CHAPTER $\text {III}$. elements of the potential theory. equation of thermal conduction.
$\text {I}$. Newtonian potential
22. Potential due to gravitating particles
23. Continuous distribution of matter
24. Theorem of total normal intensity
25. Poisson's equation. Vector potential
26. Expression of a vector function as the sum of lamellar and solenoidal components
27. Surface distribution of matter
28. Theorems on harmonic functions
$\text {II}$. Equation of Thermal Conduction
29. Fourier's law for isotropic bodies
30. Differential equation of conduction
31. A problem in heat conduction
32. Solution of the problem
Exercises

CHAPTER $\text {IV}$. motion of frictionless fluids.
33. Pressure at a point
34. Local and individual rates of change
35. Equation of continuity
36. Boundary condition
37. Eulerian equation of motion
38. Definitions: Line of flow. Vortex line. Vorticity. Circulation
39. Fluid in equilibrium
40. Equation of energy
42. Impulsive generation of motion
Irrotational Motion.
43. Velocity potential. Bernoulli's theorem
44. Simply-connected region. Velocity potential single-valued
45. Theorem due to Kelvin
46. Source, sink, or doublet in a liquid
47. Differential equation of sound propagation
Vortex Motion.
48. Vorticity or molecular rotation
49. Vortex cubes and filaments. Strength uniform
50. Kelvin's circulation theorem. Corollaries
51. Helmholtz' theorem. Vector potential
52. Kinetic energy
Exercises

CHAPTER $\text {V}$. linear vector functions. central quadric surfaces.
$\text{I}$ Dyadics
56. Scalar and vector of a dyadic
57. Products of dyadics. Distributive and associative laws
58. The cross product $\Phi \times \mathbf r$
59. The Idemfactor or Identical Dyadic $\mathrm I$
60. Reciprocal dyadics. Reciprocal of a product and of a power
62. Resolution of a dyadic into self-conjugate and anti-self-conjugate parts
63. The theorem $\mathbf r \times \mathbf a = \mathbf r \cdot \paren {\mathbf a \times \mathrm I}$
64. Simple form for any self-conjugate dyadic
65. Invariants of a self-conjugate dyadic
$\text{II}$ Central Quadric Surfaces
66. Equation of surface $\mathbf r \cdot \Phi \cdot \mathbf r =$ const.
67. Tangent plane. Perpendicular from centre
68. Condition of tangency
69. Polar plane
70. Diametral plane. Conjugate diameters. Case of ellipsoid
Exercises

CHAPTER $\text {VI}$. the rigid body. inertia dyadic. motion about a fixed point.
$\text{I}$ The Inertia Dyadic.
72. Moments and products of inertia
73. Theorem of parallel axes
74. Nonion form of the inertia dyadic
75. Momental ellipsoid and ellipsoid of gyration
76. Principal axes at any point. Binet's theorem
$\text{II}$ Motion about a Fixed Point.
77. Kinematical
78. Equation of motion
79. Motion under no forces. Poinsot's description of the motion
80. MacCullagh's description of the same
81. Impulsive forces
82. Centre of percussion for a given axis
Exercises

CHAPTER $\text {VII}$. dyadics involving $\nabla$.
83. The operator $\nabla$ applied to a vector
85. Formulæ of expansion
Transformation of Integrals.
86. Line and surface integrals
87. Surface and space integrals
Exercises

CHAPTER $\text {VIII}$. equilibrium of deformable bodies. motion of viscous fluids.
$\text{I}$ Strain Relations.
88. Homogeneous strain
89. Small homogeneous strain
90. Heterogeneous strain
91. Explicit expressions. Components of strain
$\text{II}$ Stress Relations.
92. Stress across a plane at a point
93. The stress equations of equilibrium
94. Geometrical representation of stress
$\text{III}$ Isotropic Bodies.
95. Stress-strain relations
96. The equations of equilibrium in terms of displacement
97. The strain-energy function
$\text{IV}$ Motion of Viscous Fluids.
98. Stress
99. Rate of strain
100. Relation between stress and rate of strain
101. Equasions of motion and continuity
102. Loss of kinetic energy due to viscosity
103. Vortex motion of a liquid
Exercises

CHAPTER $\text {IX}$. elementary theory of electricity and magnetism.
Intensity and Potential.
104. Point charges and poles
105. Continuous distributions
Magnetism.
106. Magnetic Moment. Short magnet
107. Two short magnets
108. Poisson's theorem of magnetisation
109. Action on a magnetized body in a non-homogeneous magnetic field
110. Magnetic induction. Permeability
111. Magnetic shell
Electrostatics.
112. Theory of dielectrics. Electric induction.
113. Coulomb's theorem
114. Boundary conditions
115. Electrical energy
Electric Currents.
116. Magnetic field associated with a current
117. Circuital theorem
118. Potential energy of a current. Mutual inductance
119. Equations of a steady electromagnetic field
120. Field due to a linear current
121. Neumann's formula for mutual inductance
122. Action of a magnetic field on a circuit carrying a current
123. Mutual action of two circuits
Exercises

CHAPTER $\text {X}$. the equations of maxwell and lorentz. the lorentz-einstein transformation.
$\text{I}$ The Electromagnetic Equations.
124. The total current. Displacement current
125. The electromagnetic equations
126. The electromagnetic potentials. Retarded or propagated potentials
127. Radiant vector or Poynting's vector
128. Electromagnetic stress and momentum
$\text{II}$ The Lorentz-Einstein Transformation.
129. Relativity in Newtonian mechanics
130. The principle o Relativity, and the Lorentz-Einstein Transformation
131. Interpretation of the transformation
132. Vectorial expression of the same
134. Transformations of $\operatorname {div} {\mathbf F}$, $\curl \mathbf F$, and $\dfrac {\partial \mathbf F} {\partial t}$
135. Transformations of the electromagnetic equations
136. Relations reciprocal. Total charge invariable
Exercises

Notation and Formulæ
Appendices
Index

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