Book:C.E. Weatherburn/Advanced Vector Analysis

C.E. Weatherburn: Advanced Vector Analysis

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

Subject Matter

• Vector Analysis

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

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