Mathematical methods. For students of physics and related fields (Q1581786)

Sometimes life brings us a pleasure of discovering, at random, a book like this. Very well written (i.e., extremely readable), very well targeted (mainly to an average student of physics at a point of just leaving his/her sophomore level) and very well concentrated (to an author's apparently beloved subject of PDE's with applications and with all their necessary pedagogically-mathematical background). The book may find a broader readership (for example via its numerous historical remarks) but seems to have been written mainly for students who wish to be given a thorough pedagogical help even at a cost of more pages to be read. For students who believe in the inseparability of physics from mathematics and (cf. the ample space for exercises) are just willing to ``do it''. There also exist some significantly weaker parts of this compendium. E.g., the absolute absence of any more abstract algebra or at least of one or two symbolic manipulation examples (transferred from the author's book on ``Mathematica''?) made me a bit dissatisfied. Figure 5.1 does not seem to offer the best explanation of the concept of sequence. Even less I did like the whole chapter on the Dirac delta function. Still, I am personally able to admit that the author of such a type of textbook has a certain right of supporting the intuitive enhancement of the accessibility of the apparatus to his readers even by occasionally offending some of those who already know. Perhaps, I was too quick by saying that the book is just about the PDE's. In fact, it covers a number of fields (cf., after all, its title) and starts by several absolutely elementary introductory chapters on calculus (starting from the chapter on coordinate systems and vectors and passing through differentiation, integration and summations till this unfortunate inset about the Dirac's trick). Then there comes the first milestone (vector analysis incl. Maxwell equations) and (only then!) the emergence of the complexified world. One only reveals the author's purpose in the subsequent returns to series (and functions) and partial differential equations. ODE's are encountered then, deduced as a special case ``from above''. Thus, chapters 12 and 13 can now move quickly through the PDE (i.e., Laplace and Schrödinger and all that) subject at last. The short final chapter 14 is (in fact, just an appendix) about nonlinearities and chaos. The main merits of the text are its clarity (achieved via returns and innovations of the context), balance (building the subject step by step) and originality (recollect: the existence of the complex numbers is only admitted far in the second half of the text!). Last but not least, the student reader is impressed by the graphical quality of the text (figures first of all, but also boxes with the essentials, summarizing comments in the left column etc) and multiple nice details (note, pars pro toto, a star in place of the dummy variable on page 82). Summarizing: Well done. A nice introduction to further reading. Not for all (some find it a bit too lengthy) but for many.