Vector Analysis and Cartesian Tensors: Third Edition

This is a comprehensive self-contained text suitable for use by undergraduate maths and science students following courses in vector analysis. It begins at an introductory level, treating vectors in terms of Cartesian components instead of using directed line segments as is often done. This novel approach simplifies the development of the basic algebraic rules of composition of vectors and the definitions of gradient, divergences and curl. The treatment avoids sophisticated definitions involving limits of integrals and is used to sustain rigorous accounts of the integral theorems of Gauss, Stokes and Green. The transition to tensor analysis is eased by the earlier approach to vectors and coverage of tensor analysis and calculus is given. A full chapter is devoted to vector applications in potential theory, including Poisson's equation and Helmholtz's theorem. For this edition, new material on the method of steepest decent has been added to give a more complete treatment, and various changes have been made in the notations used. The number and scope of worked examples and problems, complete with solutions, has been increased and the book has been redesigned to enhance the accessibility of material.

Table of Contents:
Part 1 Rectangular Cartesian co-ordinates and rotation of axes: rectangular Cartesian co-ordinates; direction cosines and direction ratios; angles between lines through the origin; the orthogonal projection of one line on another; rotation of axes; the summation convention and its use; invariance with respect to a rotation of the axes; matrix notation. Part 2 Scalar and vector algebra: scalars; vectors - basic notions; multiplication of a vector by a scalar; addition and subtraction of vectors; the unit vectors i, j, k; scalar products; vector products; the triple scalar product; the triple vector product; products of four vectors; bound vectors. Part 3 Vector functions of a real variable - differential geometry of curves: vector functions and their geometrical representation; differentiation of vectors; differentiation rules; the tangent to a curve - smooth, piecewise smooth, and simple curves; arc length; curvature and torsion; applications in kinematics. Part 4 Scalar and vector fields: regions; functions of several variables; definitions of scalar and vector fields; gradient of a scalar field; properties of gradient; the divergence and curl of a vector field; the del-operator; scalar invariant operators; useful identities; cylindrical and spherical polar co-ordinates; general orthogonal curvilinear co-ordinates; vector components in orthogonal curvilinear co-ordinates; vector analysis in n-dimensional space. Part 5 Line, surface, and volume integrals: line integral of scalar field; line integrals of a vector field; repeated integrals; double and triple integrals; surfaces; surface integrals; volume integrals. Part 6 Integral theorems: introduction; the divergence theorem (Gauss' theorem); Green's theorems; Stoke's theorem; limit definitions of div F and curl F; geometrical and physical significance of divergence and curl. Part 7 Applications in potential theory: connectivity; the scalar potential; the vector potential; Poisson's equation; Poisson's equation in vector form; Helmholtz's theorem; solid angles. Part 8 Cartesian tensors: introduction; Cartesian tensors - basic algebra; isotropic tensors; tensor fields; the divergence theorem in tensor field theory. Part 9 Representation theorems for isotropic tensor functions: introduction; diagonalization of second order symmetrical tensors; invariants of second order symmetrical tensors; representation of isotropic vector functions; isotropic scalar functions of symmetrical second order tensors; representation of an isotropic tensor function. Appendices: determinants; the chain rule for Jacobians.