An entire generation of mathematicians has grown up during the time be- tween the appearance of the first edition of this textbook and the publication of the fourth edition, a translation of which is before you. The book is famil- iar to many people, who either attended the lectures on which it is based or studied out of it, and who now teach others in universities all over the world. I am glad that it has become accessible to English-speaking readers.

This textbook consists of two parts. It is aimed primarily at university students and teachers specializing in mathematics and natural sciences, and at all those who wish to see both the rigorous mathematical theory and examples of its effective use in the solution of real problems of natural science.

Note that Archimedes, Newton, Leibniz, Euler, Gauss, Poincare, who are held in particularly high esteem by us, mathematicians, were more than mere mathematicians. They were scientists, natural philosophers. In mathematics resolving of important specific questions and development of an abstract gen- eral theory are processes as inseparable as inhaling and exhaling. Upsetting this balance leads to problems that sometimes become significant both in mathematical education and in science in general.