Vector Analysis and Cartesian Tensors, Third edition

P C Kendall, D.E. Bourne

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780748754601
• eBook ISBN: 9781351405614
• Edition: 3
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This is a comprehensive and self-contained text suitable for use by undergraduate mathematics, science and engineering students. Vectors are introduced in terms of cartesian components, making the concepts of gradient, divergent and curl particularly simple. The text is supported by copious examples and progress can be checked by completing the many problems at the end of each section. Answers are provided at the back of the book.

1 Rectangular cartesian coordinates and rotation of axes

1.1 Rectangular Cartesian Coordinates

From a fixed point O, which we shall call the origin of coordinates, draw three fixed lines Ox, Oy, Oz at right angles to each other as in Fig. 1.1. These will be called the x-axis, y-axis, z-axis respectively and they will be referred to collectively as rectangular cartesian axes Oxyz. The planes Oyz, Ozx and Oxy are called the coordinate planes and they may be referred to as the yz-plane, zx-plane and xy-plane, respectively.

Fig. 1.1 Rectangular cartesian coordinates.

It is customary to choose the axes in such a way that Ox, Oy and Oz form a right-handed set, in that order. This means that to an observer looking along Oz, the sense of the smaller arc from a point on Ox to a point on Oy is clockwise. Figure 1.2(a) illustrates this and Fig. 1.2(b) shows the relationship of Fig. 1.2(a) to a right hand. Notice that to an observer looking along Ox, the sense of the smaller arc from Oy to Oz is clockwise; and to an observer looking along Oy, the sense of the smaller arc from Oz to Ox is clockwise.

The three statements concerning observers looking along the respective axes exhibit cyclic symmetry in x, y, z, i.e. if, in any one of the three statements, we replace x by y, y by z and z by x, then one of the other two statements is produced. The operation of replacing x ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface
• Preface to second edition
• 1 Rectangular cartesian coordinates and rotation of axes
• 2 Scalar and vector algebra
• 3 Vector functions of a real variable. Differential geometry of curves
• 4 Scalar and vector fields
• 5 Line, surface and volume integrals
• 6 Integral theorems
• 7 Applications in potential theory
• 8 Cartesian tensors
• 9 Representation theorems for isotropic tensor functions
• Appendix A Determinants
• Appendix B Expressions for grad, div, curl and ∇2 in cylindrical and spherical polar coordinates
• Appendix C The chain rule for Jacobians
• Answers to exercises
• Index