Bourne D.E., Kendall P.C. Vector Analysis and Cartesian Tensors

CRC Press, 2018. — 315 p.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.Preface to second edition.
Rectangular cartesian coordinates and rotation of axes.
Scalar and vector algebra.
Vector functions of a real variable. Differential geometry. Differential geometry of curves.
Scalar and vector fields.
Line, surface and volume Integrals.
Integral theorems.
Applications in potential theory.
Cartesian tensors.
Representation theorems for isotropic tensor functions.
Determinants (App. A).
Expressions for grad, div, curl, and V 2 in cylindrical and spherical polar coordinates (App. B).
The chain rule for Jacobians (App. C).
Answers to exercises.