This revised second edition provides a systematic account of vector
algebra and calculus from the fundamental definitions. Improvements
include a rigorous proof of Taylor's theorem for a vector function, a
vectorial definition of directional derivative, more material on Green s
identity and several extra examples. The book is intended for first year
students reading an honours course in mathematics and for students of
mathematics, the physical sciences and engineering at universities and
technical colleges.
The theory is developed with emphasis on the vector as an entity in itself rather than on its definition as a number triple. It includes a chapter on line integrals, surface integrals and volume integrals, as many students require these concepts in vector theory before they have taken a formal course on integration.
The theory is presented independently of any co-ordinate system. In particular, the gradient is defined in terms of the directional derivative whilst the divergence and curl are defined by means of limits of integrals.
Author(s): Barry Spain
Edition: Second
Publisher: Van Nostrand Reinhold
Year: 1967
Language: English
Pages: 126
Preface v
Chapter 1 Vectors
1 Vectors 1
2 Addition of vectors 2
3 Subtraction of vectors 4
4 Multiplication of a vector by a scalar 5
5 Point of division 6
6 Components of a vector 8
7 Fundamental system of vectors 10
8 Scalar product 11
9 Vector product 14
10 Scalar triple product 16
11 Vector triple product 18
12 Products of four vectors 20
13 Reciprocal basis 21
Chapter 2 Applications to Space Geometry
14 Straight line 23
15 Plane 25
16 Shortest distance between two skew lines 26
Chapter 3 Differential Vector Calculus
17 Derivative of a vector 29
18 Derivative of a sum of vectors 30
19 Derivative of the product of a scalar and a vector function 30
20 Derivative of a scalar product 31
21 Derivative of a vector product 31
22 Taylor's theorem for a vector function 32
23 Derivative of a vector referred to a fundamental system 33
24 Partial derivatives of vectors 34
Chapter 4 Applications to Differential Geometry
25 Curve and tangent vector 36
26 Frenet formulae 38
27 Curvature and torsion 40
28 Surfaces and normals 43
29 Length of arc on a surface 44
30 Scalar and vector element of area 46
Chapter 5 Integration
31 Riemann integral 47
32 Line integral 47
33 Vector line integral 49
34 Double integral 50
35 Surface integral 52
36 Volume integral 55
Chapter 6 Gradient of a Scalar Function
37 Directional derivative 56
38 Gradient of a scalar function 57
39 Irrotational vector 60
40 Integral definition of gradient 63
Chapter 7 Divergence of a Vector
41 Divergence of a vector 65
42 Gauss's theorem 66
43 Divergence of the product of a scalar and a vector 68
Chapter 8 Curl of a Vector
44 Curl of a vector 70
45 Curl of the product of a scalar and a vector 71
46 Divergence of a vector product 72
47 The operator a- grad 72
48 Gradient of a scalar product 73
49 Curl of a vector product 74
Chapter 9 Stokes's Theorem
50 Alternative definition of curl 75
51 Stokes's theorem 76
52 Surface integral of the curl of a vector 78
53 Curl of the gradient of a scalar 79
54 Divergence of the curl of a vector 80
55 Solenoidal vectors 80
Chapter 10 Green's Theorems
56 Green's theorems 83
57 Harmonic functions 84
58 Uniqueness theorem 84
59 Solid angle 85
60 Green's identity 86
Chapter 11 Orthogonal Curvilinear Coordinates
61 Curvilinear coordinates 89
62 Orthogonal curvilinear coordinates 90
63 Gradient 92
64 Divergence 93
65 Curl 94
66 Curl of the curl of a vector 95
Chapter 12 Contravariance and Co variance
67 Contravariant components 97
68 Covariant components 98
69 Fundamental tensors 99
70 Natural basis 100
71 Physical components of a vector 101
72 Derivatives of natural basis vectors 102
73 Derivatives of vectors 103
74 Gradient 104
75 Divergence 105
76 Curl 106
Solutions 108
Index 112
