Description
The goal of this book is to expose the reader to the indispensable role that mathematics plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green’s functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fibre bundles and their applications to differential geometry and gauge theories.
This second edition is a substantial revision with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras, fibre bundles, and gauge theories. The spirit of the first edition, namely the balance between rigour and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the “unreasonable effectiveness of mathematics” in modern physics.
One of my favorite mathematical physics books. Enormously complete for most work.
However id not recommend for anyone looking to learn and struggle. In my opinion this is best as a text to refer to and think more about various subjects.
This product, a 1200- page encyclopedia of graduate-level mathematical physics, is a sequel to the author’s “Mathematical Methods for Students of Physics,” a book not only written with undergraduates in mind instead, but one that also carries subjects like Complex Analysis, Differential Equations and Tensors nearly seamlessly with this volume, making these two works a two-thousand page masterpiece deserving of its place by Gravitation and Classical Physics on my shelf.
Starting with finite-dimensional vector spaces (ending all the way over in the Polar Decomposition, meaning this is very nearly a self-enclosed course in applied Linear Algebra including operator theory!), the book then addresses infinite-dimensional vector spaces like Hilbert Spaces, orthogonal polynomials and everyone’s favorite Fourier approximation methods. The third subect continues the earlier volume’s Complex Analysis work, ending in advanced topics like the all-important meromorphic functions, but reviewing a bit more from the earlier volume with other topics like the Cauchy integral (while adding critical methods like the Principal Value, as well).
After continuing the Differential Equations sections from the earlier volume, capping off with a treatment of Sturm-Liouville, the volume continues with group and Lie theory before moving on to the section I’ve gone in depth on the most – tensors and differential geometry. This is a well-done section respectful of the mathematics behind the treatment. Gauges and Calculus of Variations are also included here. Although Hassani is an admitted “diffy-phile,” I think this part – from Group/Lie to Riemannian/Gauge – is the best part of this amazing tome. This will give me years of learning and re-learning.
If you are a mathematical physicist or are interested in it as a possible area of exploration as a critical link between these two regal disciplines, buy this book, and buy its prequel, Mathematcal Methods for Students of Physics. These combine to form two thousand pages of a classic in the making for this field.
The only critique – apart from occasional grammar and other typos bespotting Springer publications lately – is a heavier preference for mathematcial notation over the home team notation for physics. I suppose I could also throw in a weakness with Probability and Statistics – which is only really covered briefly at the end of the preceding volume. These are nothing to the value the book gives, however. Get both in this set!
Whatever Problem you’re facing, the book has a Chapter about it. There are Definitions with explanations of them, which is very usefull. Also Examples and problems for your own turn. I bought myself a lot of Mathbooks, some of them i regret, but i didn’t regret the buy of this one.
Un ottimo testo. Tanto vasto quanto ben scritto. Eccellente per l’insegnamento “Metodi Matematici per la Fisica” del secondo anno del corso di Laurea in Fisica.
I have owned this book for a month or so now and have been using it primarily for the algebra chapters. So far, it has exceeded my expectations.
Depth. This book is pretty thick (3″, actually). There are 37 chapters grouped into 10 parts: Finite Vector Spaces, Infinite Vector Spaces, Complex Analysis, Differential Equations, Operators, Green’s Functions, Groups/Representations, Lie Groups, and Fiber Bundles. Hassani actually renders quite a few of my other books unnecessary.
Clarity. Concepts are explained clearly and the author really tries to help you understand the abstract concepts that are presented within. Take, for instance, the section on group actions. Following five definitions thrown at you in a single paragraph, you are presented with Remark 23.3.1:
“… If you think of G_m [the stabilizer of m] as those elements of G that are confined to (stuck, or imprisoned at) m, then a “free” action of G does not allow any point of M to imprison any subset of G. …”
Referencability. Typically textbooks have to make tradeoffs between being good learning material and being a good reference. Hassani I would say does both well. While being very clear and readable, the book is also arranged very nicely for anyone looking for a reference textbook. Important definitions and theorems are boxed and key points are summarized in the margins.
In addition, each section contains historical notes about various mathematicians and physicists that are quite interesting to read. I do still have a couple of small complaints, though. The first is the size of the book. The thickness of the pages renders the book very large which makes it difficult to transport and to read on the bus. Given the range of topics covered, I would prefer the book be split into multiple volumes. Secondly, though I did mention clarity as a strength of this book, this does vary somewhat from chapter to chapter and some topics (the geometry chapters in particular, Clifford algebras especially) seem to be following a define-first-motivate-later approach.