N.-Y.: Wiley-VCH, 2014. - 634p.The intensely fruitful symbiosis between physics and mathematics is nothing short of miraculous. It is not a symmetric interaction; however, physical laws, which involve relations between variables, are always in need of rules and techniques for manipulating these relations: hence, the role of mathematics in providing tools for physics, and hence, the need for books such as this one. Physics, in its turn, supplies motivation for the development of mathematically sound techniques and concepts; it is enough to mention the importance of celestial mechanics in the evolution of perturbation methods and the impetus Dirac’s delta function gave to the work on generalized functions. So perhaps a sister volume to the present compendium would be Physical Insights forMathematicians. (From the above, one might get the impression that physics and mathematics are two separate, even opposing, domains. This is of course not true; there is a well-established field of mathematical physics and many mathematicians working in fluid and continuum mechanics would be hard-pressed to separate the mathematics and the physics elements of their seamless activity.) The present book is intended for the use of advanced undergraduates, graduate students, and researchers in physics, who are aware of the usefulness of a particular mathematical approach and need a quick point of entry into its vocabulary, main results, and the literature. However, one cannot cleanly single out parts of mathematics that are useful for physics and ones that are not; for example, while the theory of operators in Hilbert spaces is undoubtedly indispensable in quantum mechanics, an area as abstruse as category theory is becoming increasingly popular in cosmology and has found applications in developmental biology.1) Hence, the concept of mathematics useful in physics arguably covers the same area as mathematics tout court, and anyone embarking on the publication of a book such as the present one does so in the certainty that no single book can do justice to the intricate interpenetration of mathematics and physics. It is quite possible to write a second, and perhaps a third volume of an encyclopedia such as ours. Let us quickly mention significant areas that are not being covered here. These include combinatorics, deterministic chaos, fractals, nonlinear partial differential equations, and symplectic geometry. It was also felt that a separate chapter on contextfree modeling was not necessary as there is an ample literature on modeling case studies. What this book offers is an attractive mix of classical areas of applications of mathematics to physics and of areas that have only come to prominence recently.Thus, we have substantive chapters on asymptotic methods, calculus of variations, differential geometry and topology of manifolds, dynamical systems theory, functional analysis, group theory, numerical methods, partial differential equations of mathematical physics, special functions, and transform methods. All these are up-to-date surveys; for example, the chapter on asymptotic methods discusses recent renormalized group-based approaches, while the chapter on variational methods considers examples where no smooth minimizers exist. These chapters appear side by side with a decidedly modern computational take on algebraic topology and in-depth reviews of such increasingly important areas as graph and network theory, Monte Carlo simulations, stochastic differential equations, and algorithms in symbolic computation, an important complement to analytic and numerical problemsolving approaches. It is hoped that the layout of the text allows for easy cross-referencing between chapters, and that by the end of a chapter, the reader will have a clear view of the area under discussion and will be know where to go to learn more. Bon voyage!
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Princeton University Press, 2007. - 672 pages.
What can a physicist gain by studing mathematics? By gathering together everything a physicist needs to know about mathematics in one comprehensive and accessible guide, this is the question Mathematics for Physics and Physicists successfully takes on.
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N.-Y.: Springer, 2015. - 440p.
Supports learning and teaching with extended exercises at the end of every chapter including solutions
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2nd edition. — Wiley-VCH, 2009. — 538 p.
All there is to know about functional analysis, integral equations and calculus of variations in one handy volume, written for the specific needs of physicists and applied mathematicians.
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Cambridge University Press, 2007. — 1262 p.
William H. Press - Raymer Chair in Computer Sciences and Integrative Biology The University of Texas at Austin
Saul A. Teukolsky - Hans A. Bethe Professor of Physics and Astrophysics Cornell University
William T. Vetterling - Research Fellow and Director of Image Science ZINK Imaging, LLC
Brian P. Flannery - Science,...