Mathematical Physics | Mathematical physics can be understood as the study of the mathematical structures behind physics.
Mathematical Physics A Modern Introduction to it Foundations by Sadri Hassani gives a rather substantial introduction to mathematical physics. One novel feature is the short biographical accounts of the people who developed the mathematics featured in the book. |
The key to the power of this book is that it discusses the two main pillars of modern mathematical physics: functional analysis and differential geometry. The level of the presentation is aimed at beginning postgraduate students in physics or mathematics.
The book is arranged into 9 sections.
Section 0 covers the mathematical preliminaries: sets, maps, metric spaces, cardanality and mathematical induction.
Section I covers finite dimensional vector spaces: vector space & linear transformations, operator algebra, the matrix representation and spectral decomposition.
Section II then moves on to describe infinite dimensional vector spaces. Topics covered here include: Hilbert spaces, generalised functions, orthogonal polynomials and Fourier analysis.
Complex analysis is the topic of Section III. Topics in this section include: complex calculus, calculus of residues as well as more advanced ideas like meromorphic functions, analytical continuation and the gamma & beta functions.
Section IV covers differential equations: separation of variables, second order linear differential equations, complex analysis of SOLDEs and integral transforms.
Hilbert spaces are the topic of Section V. Topics covered include: basic operator theory (bounded & compact operators and their spectra), integral equations and Sturm-Liouville systems.
Section VI introduces Green’s functions in one dimension and then goes on to discuss Green’s functions in multidimensions. Both the formalism and specific examples are discussed.
After this the book takes on a more geometric direction.
Section VII covers groups and manifolds. This section covers: elementary group theory, group representations, tensor algebra, differential manifolds and tensor calculus. Also covered is exterior calculus and basics of symplectic geometry.
Section VIII covers Lie groups and their applications. Here differential geometry is developed. Topics include: Lie groups & Lie algebras, differential geometry (vector fields, Riemannian metrics, covariant derivatives, geodesics, Killing vector fields), Lie groups and differential equations and the calculus of variations. Within this section general relativity is briefly discussed as is Noether’s theorem.
The book contains exercises and plenty of worked examples.
Hardcover: 1046 pages
Publisher: Springer; 1st edition (February 8, 1999)
Language: English
ISBN-10: 0387985794
ISBN-13: 978-0387985794
I have no idea how anyone can cover all of those topics and do them justice in a single book of only about 1000 pages. I have about 1000 books to cover all of this !
@DrRocket,
Hassani does a good job at introducing the reader to these topics, but the book is by no means a full treatment. The book is great for a general overview of some of the more common mathematical threads of physics, but any serious reader needs to employ more targeted books to get a true understanding.