Principles and Techniques of Applied Mathematics

Principles and Techniques of Applied Mathematics

Principles and Techniques of Applied Mathematics

Principles and Techniques of Applied Mathematics

Excerpt

For many years the gap between pure mathematics and applied mathematics has steadily widened. On the one hand, the pure mathematicians are considering structures and systems which are becoming ever more abstract and general; on the other hand, the applied mathematicians are studying concrete and specific problems. It is well known, of course, that this gap between the two groups is really illusory, that the study of abstract systems can help in the solution of concrete problems, and that the study of specific problems may suggest interesting generalizations for the pure mathematicians.

This book was written in an attempt to show how the powerful methods developed by the abstract studies can be used to systematize the methods and techniques for solving problems in applied mathematics. Such a systematic treatment requires a great deal of preparation by the student; consequently, more than half of the book is devoted to a study of abstract linear spaces and of operators defined on such spaces. However, in this treatment, the emphasis is not on the abstract theory but on the techniques which can be derived from this theory to solve specific problems. For example, Chapter 3 presents the elements of Laurent Schwartz's "Theory of Distributions" in a form which should make it more accessible to the people who would use it--the applied mathematicians, the physicists, and the engineers.

An introductory book on applied mathematics, such as this one, can by its very nature contain little that is new or original. However, it is believed that some of the techniques presented here, such as those for solving integral equations, for finding the Green's function for ordinary or partial differential equations, and for finding the spectral representation of ordinary differential operators, may be relatively unfamiliar to the general reader. The development and exposition of these techniques are the main purpose of this book.

As far as possible, I have attempted to present the subject so as to lay stress upon the ideas and not upon the minutiae of the proofs; consequently, many details, illustrations, and extensions of the text have been put into problems and appendices. It is recommended that the reader study the problems as well as the text in order to get a more complete knowledge of the subject.

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