Mathematical Physics: A Modem Introduction to Its Foundations

by Sadri Hassani

Organization and Topical Coverage
Aside from Chapter 0, which is a collection of pnrely mathematical concepts,
the book is divided into eight parts. Part I, consisting of the first fonr chapters, is
devoted to a thorough study of finite-dimensional vectorspaces and linear operators
defined on them. As the unifying theme ofthe book, vector spaces demand careful
analysis, and Part I provides this in the more accessible setting of finite dimension in
a language thatis conveniently generalized to the more relevant infinite dimensions,'
the subject of the next part.
Following a brief discussion of the technical difficulties associated with in-
finity, Part IT is devoted to the two main infinite-dimensional vector spaces of
mathematical the classical orthogonal polynomials, and Foutier series
and transform.
Complex variables appear in Part ill. Chapter 9 deals with basic properties of
complex functions, complex series, and their convergence. Chapter 10 discusses
the calculus of residues and its application to the evaluation of definite integrals.
Chapter II deals with more advanced topics such asmultivalued functions, analytic
continuation, and the method of steepest descent.
Part IV treats mainly ordinary differential equations. Chapter 12 shows how
ordinary differential equations of second order arise in physical problems, and
Chapter 13 consists of a formal discussion ofthese differential equations as well
as methods of solving them numerically. Chapter 14 brings in the power of com-
plex analysis to a treatment of the hypergeometric differential equation. The last
chapter of this part deals with the solution of differential equations using integral
transforms.
Part V starts with a formal chapter on the theory of operator and their spectral
decomposition in Chapter 16. Chapter 17 focuses on a specific type of operator,
namely the integral operators and their corresponding integral equations. The for-
malism and applications of Stnrm-Liouville theory appear in Chapters 18 and 19,
respectively.
The entire Part VI is devoted to a discussion of Green's functions. Chapter
20 introduces these functions for ordinary differential equations, while Chapters
21 and 22 discuss the Green's functions in an m-dimensional Euclidean space.
Some of the derivations in these last two chapters are new and, as far as I know,
unavailable anywhere else.
Parts VII and vrncontain a thorough discussion of Lie groups and their ap-
plications. The concept of group is introduced in Chapter 23. The theory of group
representation, with an eye on its application in quantom mechanics, is discussed
in the next chapter. Chapters 25 and 26 concentrate on tensor algebra and ten-,
sor analysis on manifolds. In Part vrn, the concepts of group and manifold are
brought together in the coutext of Lie groups. Chapter 27 discusses Lie groups
and their algebras as well as their represeutations, with special emphasis on their
application in physics. Chapter 28 is on differential geometry including a brief
introduction to general relativity. Lie's original motivation for constructing the
groups that bear his name is discussed in Chapter 29 in the context of a systematic
treatment of differential equations using their symmetry groups. The book ends in
a chapter that blends many of the ideas developed throughout the previous parts
in order to treat variational problems and their symmetries. It also provides a most
fitting example of the claim made at the beginning of this preface and one of the
most beautiful results of mathematical Noether's theorem ou the relation
between symmetries and conservation laws.

2273 pages, Kindle Edition

Published February 25, 2018