Ebook: Mathematical Handbook: Higher Mathematics

About the book

This handbook is a continuation of the Handbook of Elementary Mathematics by the same author and includes material usually studied in mathematics courses of higher educational institutions.
The designation of this handbook is two fold.

Firstly, it is a reference work in which the reader can find definitions (what is a vector product?) and factual information, such as how to find the surface of a solid of revolution or how to expand a function in a trigonometric series, and so on. Definitions, theorems, rules and formulas (accompanied by examples and practical hints) are readily found by reference to the comprehensive index or table of contents.

Secondly, the handbook is intended for systematic reading. It does not take the place of a textbook and so full proofs are only given in exceptional cases. However, it can well serve as material for a first acquaintance with the subject. For this purpose, detailed explanations are given of basic concepts, such as that of a scalar product (Sec. 104), limit (Secs. 203~206), the differential (Secs. 228-235), or infinite series (Secs. 270, 366-370). All rules are abundantly illustrated with examples, which form an integral part of the hand­book (see Secs. 50-62, 134, 149, 264-266, 369, 422, 498, and others). Explanations indicate how to proceed when a rule ceases to be valid; they also point out errors to be avoided (see Secs. 290, 339, 340, 379, and others).

The theorems and rules are also accompanied by a wide range of explanatory material. In some cases, emphasis is placed on bringing out the content of a theorem to facilitate a grasp of the proof. At other times, special examples are illustrated and the reasoning is such as to provide a complete proof of the theorem if applied to the general case (see Secs. 148, 149, 369, 374). Occasionally, the explanation simply refers the reader to the sections on which the proof is based. Material given in small print may be omitted in a first read­ing however, this does not mean it is not important.

Considerable attention has been paid to the historical background of mathematical entities, their origin and development. This very often helps the user to place the subject matter in its proper perspective. Of particular interest in this respect are Secs. 270, 366 together with Secs. 271, 383, 399, and 400, which, it is hoped, will give the reader a clearer understanding of Taylor’s series than is usually obtainable in a formal exposition. Also, biographical information from the lives of mathematicians has been included where deemed advisable.

The book was translated from Russian by George Yankovsky was published in 1987 (fifth reprint) by Mir Publishers.