This book deals with vector algebra and analysis and with their application to three-dimensional geometry and the analysis of fields in three dimensions. While many treatments of the application of vectors have approached the fundamentals of the subject intuitively, assuming some prior knowledge of Euclidean and Cartesian geometry, Professor Chrisholm here bases the subject on the axioms of linear space algebra, which are fundamental to many branches of mathematics. While developing the properties of vectors from axioms, however, he continually emphasizes the geometrical interpretation of vector algebra in order to build up intuitive relations between the algebraic equations and geometrical concepts. Throughout, examples are used to illustrate the theory being developed; several sets of problems are incorporate in each chapter, and outline answers to many of these are given. Written primarily for undergraduate mathematicians in the early part of their courses, this lucidly written book will also appeal to mathematical physicists and to mathematically inclined engineers.
Author(s): J. S. R. Chisholm
Publisher: CUP
Year: 1978
Language: English
Pages: 305
CONTENTS......Page 5
Preface......Page 7
1.1 Introduction......Page 13
1.2 Scalar multiplication of vectors......Page 15
1.3 Addition and subtraction of vectors......Page 17
1.4 Displacements in Euclidean space......Page 24
1.5 Geometrical applications......Page 29
2.1 Scalar products......Page 36
2.2 Linear dependence and dimension......Page 42
2.3 Components of a vector......Page 47
2.4 Geometrical applications......Page 54
2.5 Coordinate systems......Page 63
3.1 The vector product of two vectors......Page 69
3.2 The distributive law for vector products; components......Page 72
3.3 Products of more than two vectors......Page 77
3.4 Further geometry of planes and lines......Page 84
3.5 Vector equations......Page 92
3.6 Spherical trigonometry......Page 96
4.1 Vectors and matrices......Page 99
4.2 Determinants; inverse of a square matrix......Page 104
4.3 Rotations and reflections in a plane......Page 114
4.4 Rotations and reflections in 3-space......Page 123
4.5 Vector products and axial vectors......Page 134
4.6 Tensors in 3-space......Page 136
4.7 General linear transformations......Page 140
5.1 Definition of curves and surfaces......Page 147
5.2 Differentiation of vectors; moving axes......Page 161
5.3 Differential geometry of curves......Page 170
5.4 Surface integrals......Page 184
5.5 Volume integrals......Page 200
5.6 Properties of Jacobians......Page 212
6.1 Scalar and vector fields......Page 215
6.2 Divergence of a vector field......Page 223
6.3 Gradient of a scalar field; conservative fields......Page 233
6.4 Curl of a vector field; Stokes' theorem......Page 242
6.5 Field operators; the Laplacian......Page 254
Appendix A Some properties of functions of two variables......Page 265
Appendix B Proof of Stokes' theorem......Page 275
Reference list......Page 281
Outline solutions to selected problems......Page 283
Index......Page 297
