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

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## C.E. Weatherburn:

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

Published $\text {1921}$, **G. Bell and Sons, Ltd.**.

### Subject Matter

### Contents

- Preface
- Historical Introduction
- Short Courses

- CHAPTER $\text {I}$. addition and subtraction of vectors. centroids.
- 1. Scalar and vector quantities
- 2. Length vectors
- 3. Definitions of terms
- 4. Addition and subtraction of vectors. Component and Resultant
- 5. Multiplication by a number
- 6. Resolution of a vector
- 7. The unit vectors $\mathbf i$, $\mathbf j$, $\mathbf k$
- 8. Division of a line in a given ratio
- 9. Centroid, or centre of mean position
- 10. Centroids of area and volume
- 11. Centre of mass
- 12. Relative position. Relative displacement. Uniform relative velocity
- 13. Concurrent forces. Vector polygon. Lami's Theorem
- 14. Soliution of examples
- Exercises

- CHAPTER $\text {II}$. elementary geometrical illustrations and applications.
- 15. Introductory
- 16. Vector equation of a straight line
- 17. Bisector of the angle between two straight lines
- 18. The Triangle
- 19. The Tetrahedron
- 20-1 Vector equation of a plane
- 22. Linear relation independent of the origin
- 23. Vector areas. Theorem for closed polyhedra
- Exercises

- CHAPTER $\text {III}$. products of two vectors. the plane and the sphere.
*Scalar and Vector Products.*- 24. New use of terms
*product*and*multiplication* - 25. Scalar product of two vectors
- 26. The distributive law
- 27. Vector product of two vectors
- 28. The distributive law

- 24. New use of terms

*Geometry of the Plane.*- 29. Vector equation of a plane $\mathbf r \cdot \mathbf n = q$
- 30. Distance of a point from a plane
- 31. Plane through the intersection of two planes
- 32. Distance of a point from a straight line

*Geometry of the Sphere.*- 33. Vector equation of a sphere
- 34. Equation of the tangent plane at a point
- 35. Polar plane of a point
- 36. Diametral plane for parallel chords
- 37-8. Radical plane of two spheres. System of spheres with a common radical plane

*Applicaiton to Mechanics*- 39. Work done by a force
- 40. Vector moment or
*torque*of a force about a point - 41. Angular velocity of a rigid body about a fixed axis
- Exercises

- CHAPTER $\text {IV}$. products of three or four vectors. non-intersecting straight lines.
*Triple and Quadruple Products*- 42. Triple products
- 43. Scalar triple product. Volume of parallelepiped. Coplanar vectors
- 44. Vector triple product
- 45. A scalar product of four vectors
- 46. A vector product of four vectors. Relation between four vectors
- 47. System of vectors reciprocal to $\mathbf {a, b, c}$

*Further Geometry of the Plane and Straight Line.*- 48. Planes satisfying various conditions
- 49. Condition of intersection of two straight lines
- 50. The common perpendicular to two non-intersecting straight lines
- 51. Plücker's coordinates of a straight line
- 52. Volume of a tetrahedron

*Other Applications.*- 53. Two formulae of Spherical Trigonometry
- 54. Rankine's theorem for four concurrent forces
- Exercises

- CHAPTER $\text {V}$. differentiation and integration of vectors. curvature and torsion of curves.
*Differentiation and Integration*- 55. Derivative of a vector with respect to a scalar variable. Differentiation
- 56. Derivatives of products
- 57. Integration of vectors and products of vectors

*Curvature and Torsion.*- 58. Tangent to a curve at a given point
- 59. Curvature. Principal normal. Plane of curvature or osculating plane
- 60. Binormal. Torsion

*Definite Integrals.*- 61. Definite integral of a vector function
- 62. Illustrations
- 63. Line integral of a vector function
- 64. Surface integral of a vector function
- Exercises

- CHAPTER $\text {VI}$. kinematics and dynamics of a particle.
*Kinematics.*- 65. Velocity at an instant. Theorem of vector addition of velocities
- 66. Acceleration at an instant. Theorem of vector addition of accelerations
- 67. Tangential and normal residues of acceleration
- 68. Radial and transverse residues of velocity and acceleration
- 69. Areal velocity about a point
- 70. Motion with constant acceleration

*Dynamics.*- 71. Momentum
- 72. Newton's Second Law of Motion
- 73. Impulse of a force. Impulsive forces
- 74. Activity of a force
- 75. The principle of energy
- 76. Moment of momentum or angular momentum about a point. Principle of A.M.
- 77. Central forces
- 78. Central forces varying inversely as the square of the distance
- 79. Planetary motion
- 80. Central force varying directly as the distance
- 81. Motion of a particle on a fixed curve
- Exercises

- CHAPTER $\text {VII}$. dynamics of a system of particles and of a rigid body.
*Dynamics of a System.*- 82. Linear momentum of the system
- 83. Acceleration of the centre of mass
- 84. Angular momentum about a point
- 85. Moving origin of moments. Centre of mass as origin
- 86. Equations for impulsive forces

*Kinematics of a Rigid Body*- 87. Motion of a rigid body about a fixed point. Instantaneous axis of rotation
- 88. General motion of a rigid body. Screw motion
- 89. Simultaneous motions

*Dynamics of a Rigid Body*- 90. Angular momentum of a rigid body. Moments and products of inertia
- 91. Principal axes of inertia
- 92. Kinetic energy of a rigid body
- 93. Principle of energy
- 94. Moving axes or frame of reference
- 95. Coriolis' Theorem
- 96. Euler's dynamical equation
- Exercises

- CHAPTER $\text {VIII}$. statics of a rigid body.
- 97. Conditions of equilibrium of a rigid body
- 98. Equivalent systems of forces
- 99. Parallel forces. Centre of gravity
- 100. Couples. Composition of couples.
- 101. Poinsot's reduction of a system of forces. Central axis. Equivalent wrench
- 102. Null plane at a point
- 103. Conjugate forces
- 104. Principle of Virtual Work, or Virtual Velocities
- 105. Equilibrium of a string under any forces
- 106. Equilibrium of a wire under any forces
- Exercises

- Summary
- Answers to Exercises
- Index

## Source work progress

- 1921: C.E. Weatherburn:
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