This remarkable book is the revised translation of the German edition published in 1996. On almost 1300 pages, Eberhard Zeidler offers a fascinating panoramic overview of mathematics, ranging from elementary results to advanced and sophisticated parts of contemporary mathematics. The book is a beautiful illustration of the fact that mathematics is much more than a dry collection of formulas, definitions, theorems and manipulation with symbols. The historical background of results and theories is explained in many places throughout the book and an emphasis to significant applications is given. The introductory chapter is a 200-page reference book on basic mathematical notions usually required by students, scientists and other practitioners. The following three chapters are devoted to analysis (375 pages), algebra (125 pages) and geometry (150 pages). A short chapter on logic and set theory follows this. The last three chapters are devoted to the following fields of applications of mathematics: calculus of variations and optimization, stochastic calculus, numerical mathematics and scientific computing. The eight chapters are divided into 62 sections and 367 subsections. More than 20 pages at the end of the book are devoted to a detailed sketch of the history of mathematics. Throughout the book, there are many tables, illustrations and indications on software systems making it possible to carry out many routine jobs in mathematics on a standard PC. Also, a rich bibliography is included.
In order to show that the book is by no means a dry collection of mathematical facts, a selection (necessarily limited) of several subtitles can be offered: the perihelion motion of Mercury, fast computers and the death of the sun, mathematics and computers – a revolution in mathematics, rigorous justifications of the Cartan differential calculus and its applications, vector analysis and physical fields, conservation laws in mechanics, applications of ODE’s to electrical circuits or chemical reactions, the two body problem, laws of Kepler, shock waves and the conditions for entropy of Lax, the Hamilton-Jacobi equations, applications to geometric optics, electrostatics and Green’s functions, applications to quantum mechanics, dynamics of gases, sound waves, applications to hydromechanics, number theory and coding theory, A. Weil and Fermat’s last theorem, the Dirac equation and relativistic electrons, spin geometry and fermions, the necessity of proofs in the age of computers, wavelets, data compression and adaptivity, etc. The book is aimed at a wide readership: students of mathematics, engineering, natural sciences, and economy, practitioners who work in these fields, school and university teachers. No doubt professional mathematicians will also find the book very useful. This fascinating book can be strongly recommended to anybody who applies mathematics or simply wants to understand important concepts and results from both classical and modern mathematics.