# Book:C.E. Weatherburn/Advanced Vector Analysis

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## C.E. Weatherburn:

## C.E. Weatherburn: *Advanced Vector Analysis*

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

### Subject Matter

### 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
- 41. Steady motion
- 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*- 53. Linear vector function. Dyadic. Dyads. Antecedents and consequents
- 54. Dyadics. Scalar multiplication. Prefactor and postfactor. Conjugate dyadics
- 55. Distributive law for dyads. Nonion form of a dyadic. Theorem
- 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
- 61. Conjugate dyadics. Conjugate of a product. Self-conjugate and anti-self-conjugate dyadics
- 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
- 71. Reciprocal quadric surfaces
- 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
- 84. Differentiation of dyadics
- 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
- 133. Addition of velocities
- 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

## Source work progress

- 1924: C.E. Weatherburn:
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