Introduction to Vector and Tensor Analysis

by Robert C. Wrede

This broad introduction to vector and tensor analysis is designed for the advanced undergraduate or graduate student in mathematics, physics, and engineering as well as for the practicing engineer or physicist who needs a theoretical understanding of these essential mathematical tools. In recent years, the vector approach has found its way even into writings on aspects of biology, economics, and other sciences.The many and various topics covered the algebra of vectors — linear dependence and independence, transformation equations, the inner product, the cross product, and the algebra of matrixes; the differentiation of vectors — geometry of space curves, kinematics, moving frames of reference, Newtonian orbits and special relativity theory; partial differentiation of vectors — geometry of space curves, kinematics, moving frames of reference, Newtonian orbits and special relativity theory; partial differentiation and associated concepts — surface representations, bases in general coordinate systems, and maxima and minima of functions of two variables; the integration of vectors — line integrals, surface integrals, surface tensors and volume integrals; tensor algebra and analysis — fundamental notions of  n-space, transformations and tensors, Riemannian geometry, tensor processes of differentiation, geodesics, the curvature tensor and its algebraic properties, and general relativity theory.Throughout, Professor Wrede stresses the interrelationships between algebra and geometry, and moves frequently from one to the other. As he points out, vector and tensor analysis provides a kind of bridge between elementary aspects of linear algebra, geometry and analysis. He uses the classical notation for vector analysis, but introduces a more appropriate new notation for tensors, which he correlates with the common vector notation. He stresses proofs and concludes each section with a set of problems designed to help the student get a solid grasp of the ideas, and explore them more thoroughly on his own. His approach features a combination of important historical material with up-to-date developments in both fields. The knowledge of vector and tensor analysis gained in this way is excellent preparation for further studies in differential geometry, applied mathematics, and theoretical physics.

Genres: Mathematics, Physics

845 pages, Kindle Edition

First published January 1, 1972

Reviews

[Paul · Rating 2 out of 5]

So I picked up this book back in the 1990s to continue my self-training in mathematics, and between my active video game addiction and, as I now see, the book itself, I didn't get very far. As it turns out, reading Penrose's Road to Reality provided considerable motivation to tackle even the chunks that I did (chapter 1 and most of chapter 5 of this 5 chapter book; definitely a book's worth of effort!); I can't imagine my teenage self having any idea why we were talking about this covariant and contravariant gibberish, for instance.

In any case, this book has serious flaws. It's presented in the formal axiomatic style of Definition This and Theorem That, but it's seriously out of order. Important terms are introduced and left undefined, sometimes for chapters, possibly never at all (if the index is to be believed)...the idiosyncratic use of the term "arrow," for example. When definitions do occur, they are by no means always clear, nor are they used coherently in the ensuing discussion. The proofs are generally pretty hurried. Without having taken the time, or having the background, to evaluate them, I merely comment that I suspect a number of them are quite shaky.

I anticipate seeking out other books to attack this material.