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

## C.E. Weatherburn: Elementary Vector Analysis

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

### 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. Solution 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
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.
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