Grinfeld M. (ed.) Mathematical Tools for Physicists

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!