Tensors, Differential Forms, and Variational Principles [antikvár]

The objective of this book is twofold. Firstly, it is our aim to present a self-contained, reasonably modern account of tensor analysis and the calculus of exterior differential forms, adapted to the needs of physicists, engineers, and applied mathematicians in general. Secondly, however, it is anticipated that a substantial part of the material included in the later chapters is of interest also to those who have some previous knowledge of tensors and differential forms: we refer in particular to the remarkable interaction between the concept of invariance and the calculus of variations, which has profound implications in almost all physical field theories. The general approach and organization of the opening chapters is determined almost exclusively by the requirements of our first objective. In fact, these chiipters are based on courses presented during the past decade at several uaiversities to audiences with widely varying interests and academic backgioi.nds. Accordingly the prerequisites consist merely of basic linear algebr;i, advanced calculus of several real variables, and some very elementary aspects of the theory of differential equations; the first five chapters therefore constitute a one-semester course which should be readily accessible to senior undergraduates majoring in mathematics, physics, or some branch of engineering. Initially the pace is very gradual, if not leisurely, with emphasis on motivation with the aid of simple physical examples, while at the same time a systematic effort is made to proceed to the core of the subject matter by way of successive abstraction from concrete situations. In this respect, therefore, our treatment does not follow the historical development of the calculus of tensors and forms, whose origins are deeply rooted in differential geometry, which does, in fact, provide their most natural setting. However, since the applications of tensors and forms have meanwhile spread to entirely different areas, the basic approach presented below has intentionally been divested of many of the customary trappings of metric differential geometry. The later parts of the book, beginning with Chapter 6, are devoted primarily to the second of the aforementioned objectives. It had been observed repeatedly by David Hilbert that the effects of invariance postulates on