Vector and Tensor Analysis [antikvár]

PREFACE The development of techniques raises more and more mathematical problems for the engineer. The mathematical formulation of technical problems is mainly the task of the engineer, but it is the mathematician who helps him to solve them. The present book is primarily for engineers. Its purpose, therefore, is much less to solve numerical problems than to present concepts and theorems clearly and accurately and to determine precisely the compass and limits of their possible practical applications. The examples have been chosen accordingly. The theorems are formulated to suit the problems actually occurring in engineering practice. The proofs — especially the reasoning used in them — are indispensable for the engineer, because it is only by their help that he can get a clear view of the content of the theorems and of the limits of their sphere of validity. The material of the first chapter, vector algebra, is certain to be known to the reader. This chapter is revisionary, with many numerical examples. The second and third chapters deal with the theory of vector-scalar and scalar-vector functions. Among the examples the number of exercises requiring numerical calculation gradually decreases for the sake of those developing theoretical considerations. Concerning the material of these two chapters we have had to assume a knowledge of the essential concepts of the differential and integral calculus and of functions with one and several variables. Nevertheless, we have included a brief recapitulation of this in the fourth chapter. The fourth and fifth chapters deal with vector-vector functions. In Chapter Four we define the most important concepts of vector analysis by means of the integral concept after Ignatowsky. Thus we get a very intuitive picture of them, but their analysis would be difficult to survey. Therefore in Chapter Five we define these notions by means of the differential quotient as well.