This book is written by a mathematician who is famous both for his fundamental contributions to modern mathematics and as an author of several textbooks and monographs on Mathematical Methods in Classical Mechanics, Ordinary Differential Equations, Singularities of Differentiable Maps, and Topological Methods in Hydrodynamics. The book is based on a short course of lectures delivered to the third year mathematics students of the Independent University of Moscow in the fall semester of 1994.
This introduction to PDEs is highly unusual and is very different from numerous textbooks on the subject. The book is not designed to serve as a systematic textbook for, e.g., science and engineering students. The reason is twofold. First, the reader is assumed to have a background in abstract differential geometry. Namely, such notions as differential forms,jets, symplectic and contact structures, cotangent bundle, Riemannian manifold, projective space etc. are used from the very beginning of the text. (Some of these notions are briefly defined but the reader should definitely be familiar with them.) Secondly, only a number of selected topics is discussed. For example, the wave equation and separation of variables are considered only in the case of a one -dimensional vibrating string. The heat equation is not considered at all. Some topics, e.g., boundary value problems for the Laplace equation, are described in a short verbal paragraphs. In fact, several topics whose description in standard texts involves a lot of important formulas are described in short verbal paragraphs. These descriptions are always very precise and go directly to the heart of the topic. However, the reader must definitely have sufficient mathematical maturity and technical skills to translate these short explanations to the language of formulas.
The topics that are selected for discussion are treated with great elegance and in a nonstandard manner, emphasizing both physical intuition and geometric insight. Some parts of the lectures contain material whose accessible exposition can hardly be found in any any other place. An example of this is an appendix devoted to topological content of the multifield representation of spherical functions (i. e., the fact that spherical functions can be obtained by calculating all partial derivatives of the Coulomb potential). The book contains many nontrivial problems.
The book can serve as a nonstandard, geometrically motivated introduction to PDEs for students having some background in pure abstract mathematics. It will also be of definite interest to specialists since it provides a fresh unusual approach to some standard topics. It is, probably, worth mentioning that the introduction contains some general philosophical views of the author on the subject of PDEs and modern mathematics as a whole and will be of interest to a broad mathematical audience.