Book:C.E. Weatherburn/Advanced Vector Analysis

Subject Matter

 * Vector Analysis

Contents

 * (''September $1923$)


 * CHAPTER $\text {I}$..
 * 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 $\nabla^2$


 * CHAPTER $\text {II}$..
 * 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


 * CHAPTER $\text {III}$..
 * $\text {I}$. 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


 * CHAPTER $\text {IV}$..
 * 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


 * CHAPTER $\text {V}$..
 * $\text{I}$ 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


 * CHAPTER $\text {VI}$..
 * $\text{I}$ 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


 * CHAPTER $\text {VII}$..
 * 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


 * CHAPTER $\text {VIII}$..
 * $\text{I}$ 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


 * CHAPTER $\text {IX}$..
 * 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


 * CHAPTER $\text {X}$..
 * $\text{I}$ 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