Geometrical Methods in The The

Newton‘s fundamental discovery, the one which he considered necessary to keep secret and published only in the form of an anagram, consists of the following: Data aequatione quotcunque jluentes quantitae involvente jluxiones invenire et vice versa. In contemporary mathematical language, this means: "It is useful to solve differential equations". At present, the theory of differential equations represents a vast con­ glomerate of a great many ideas and methods of different nature, very useful for many applications and constantly stimulating theoretical in­ vestigations in all areas of mathematics. Many of the routes connecting abstract mathematical theories to appli­ cations in the natural sciences lead through differential equations. Many topics of the theory of differential equations grew so much that they became disciplines in themselves; problems from the theory of differential equations had great significance in the origins of such disciplines as linear algebra, the theory of Lie groups, functional analysis, quantum mechanics, etc. Consequently, differential equations lie at the basis of scientific mathematical philosophy (Weltanschauung). In the selection of material for this book, the author intended to expound basic ideas and methods applicable to the study of differential equations. Special efforts were made to keep the basic ideas (which are, as a rule, simple and intuitive) free from technical details. The most fundamental and simple questions are considered in the greatest detail, whereas the exposition of the more special and difficult parts of the theory has been given the character of a survey.