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Lectures on partial differential equations. (Vorlesungen über partielle Differentialgleichungen. Übersetzt aus dem Russischen von Tobias Damm.) (German) Zbl 1076.35001
Springer-Lehrbuch. Berlin: Springer (ISBN 3-540-43578-6/pbk). ix, 174 S. (2004).
This book provides an introductory text (in German) to basic partial differential equations, based on the author’s lectures at Moscow University [Moscow: Fazis (1997; Zbl 1055.35500)]. Most of the standard themes are treated (see list below), but some unusual topics are covered as well. For instance, in chapter 10 double layer potentials are considered, and chapters 11 and 13 deal (among others) with Maxwell’s theorem on the multipole expansion of spherical functions. The style of the book is quite non-technical (it contains almost no estimates), taking a mainly geometric viewpoint. It might well be that undergraduate students could encounter problems already with Exercise 1 on p. 1 and the proof of Theorem 1 on p. 2. A key word index would come as a welcome addition to the book.
The chapters are: 1.General theory of first order equations 2. Continuation of 1. 3. Huygens’ principle and the spreading of waves 4. The string (d’Alembert’s method) 5. Fourier’s method for a string; 6. Theory of oscillations and the variational principle 7. Continuation of 6. 8. Properties of harmonic functions 9. Fundamental solutions of the Laplace operator and potentials 10. The double layer potential 11. Spherical harmonics, Maxwell’s theorem and removable singularities 12. Boundary value problems for the Laplace equations and the theory of linear equations and systems 13. The topological content of Maxwell’s theorem concerning the multipole expansion of spherical functions 14. Appendix 2: Exercises

MSC:
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
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