Customer Reviews

Mathematical Physics

5 star:		(6)
4 star:		(3)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

I agree with other reviewers that this book is the first choice if you want to get a handle on mathematical methods of theoretical physics at advanced undergraduate / beginning graduate level. The nearest competitor is Byron & Fuller's "Mathematics of Classical and Quantum Physics" which has been around a long time and has many good points; but having used both I prefer this. The level and philosophy is about the same but the coverage is wider and the presentation clearer and cleaner. It's a pleasure to read.

The book is divided into eight parts, each comprising three or four chapters, on: Finite-dimensional Vector Spaces, Infinite-dimensional Vector Spaces, Complex Analysis, Differential Equations, Operators on Hilbert Spaces, Green's Functions, Groups and Manifolds, Lie Groups and Applications. Fear not: although it isn't designed for freshmen, it emphatically isn't the sort of math book where you have to crack the code to get any benefit.

The layout is excellent, there are many, many worked examples, and I found very few slips or typos. One black mark, the reason I don't give it 5 stars: although there are a massive 850 problems, there are no solutions (just like Byron & Fuller). Unless you're confident in your mathematical ability, you may find that a drawback for self-study. Finally, a word to the wise: check out this title at amazon.co.uk (provided you aren't in a hurry).

I share the excitement in former reviewers` comments on the overall quality of this book. However, it is important to notice that Hassani does not try to present the material covered in exhaustive detail. This is, however, by no means a criticism given the breadth of topics pertinent to the study of mathematical physics. Therefore, it is impossible to distill such a plethora of material while retaining the full rigor.

The book structures itself around the concept of a vector space, and the author does not shy away from abstractions which involve mathematical structures such as fields, algebras, groups, etc. as well as the topological concepts like completeness, compactness, bounded operators and so on. In this respect, the view toward math is modern and stresses the mutualism between physics and math in the advancement of both.

In general, there is a noticeable trend shift in writing of mathematical physics texts, which was inaugurated by Dennery and Krzywicki`s text and Walter Thirring`s two volume classic as opposed to the Morse and Feshbach variety which mostly focuses on the detailed solutions of certain problems of interest in physics. In this respect, this book is a nice complement and update to both. The quality of writing is reflected on the references and again culling the best of two worlds. Indeed, the references span numerous important fields and approaches to math (Rudin`s Functional Analysis, Bott & Tu`s Differential Forms in Algebraic Topology, Barut & Raczka`s Group Representation Theory...), and this more than compensates the absence or truncation of any ideas/concepts.

The topics that are absent from the text include measure theory, a possible prologue to algebraic geometry, hyper-complex analysis and geometric algebra (except for a short digression on quaternions). However, this does not devalue the book owing to the reasons presented above. I highly recommend this text for anyone who has an appreciation of the strong link between the physical world and its description via mathematical constructions.

I used this book in a graduate course I recently attended in mathematical physics. At the beginning of the course, I wrestled with this book's treatment of introductory topics such as finite-dimensional vector spaces and operator theory. I struggled with unfamiliar notation and what I felt was a lack of concrete examples. As the course progressed into infinite-dimensional vector spaces however, the unfamiliar notation began to make more sense and I started to appreciate its compactness. I began to understand that the lack of concrete examples was brevity for the sake of completeness. By the end of the course, Hassani's book had become my desktop reference for mathematical physics. The treatment of Green's functions is particularly strong, taking up three solid chapters. Compare this to Arfken's "Mathematical Methods for Physicist" where Green's functions are given only a single subsection. Not to detract from Arfken's book, I just found his coverage of this topic to be weak.

Given all of this, I highly recommend a supplementary text for those using this book to learn new material - Arfken and "Mathematics of Classic and Quantum Physics" by Byron and Fuller are both very readable. Those using it as a refresher or reference will find a thorough, compact, and consistent book.

Excellent. The book covers almost everything you need to know in a clear, logical and most importantly for physicists, applied manner. The choice of examples used in the many, many worked problems in the main body of the text is extremely clever, particularly if you are interested in gaining a working facility with quantum mechanics. They serve to illustrate very clearly the links between seemingly (particularly if you have tacked together a similar body of knowledge from a host of smaller books aimed a mathematicians) unrelated areas, giving the book a very cohesive feel. The only let-down (which is alleviated by the many worked examples) is the lack of answers to the problem sets. However, given the amount of material the book covers, if the student were to supplement it with a Schaum's Outline or two, they would have absolutely everything they need to become (more than) competent. The layout and type-setting are also superb, and the short biographies included are a welcome addition, making the book feel slightly less formal, which I found a breath of fress air in comparison to other texts on the subject. In short, a must have.

Up-to-date, thorough, clear and reasonably rigorous, this volume sets a new standard in mathematical physics textbooks. It is so good that the reader can only hope for a few more pages (path integrals, topology) in a new edition or another book.

Students plod through Applied Math graduate courses with desk tops groaning under many supplementary texts - linear algebra, Functional Analysis and some Upper-level Physics/ Electrical Engineering books. Learning a new topic often involves frantic hunting through numerous texts to refresh one's memory of the underlying concepts. The primary strength of Hassani's book is to spare the reader of this "where is Waldo?" tedium.

Hassani first introduces the concept of a vector space and gives numerous examples including the less "intuitive" function spaces and matrix spaces. He quickly builds upon this idea to encompass linear operators, algebras and functions defined in terms of them. Hassani's sweep of basic concepts is comprehensive and thorough while seamlessly weaving in ideas from many different branches of mathematics that provide the edifice for much of modern physics. The side notes enable one to browse through the book and find a particular topic. Interspersing the text with short biographical sketches of mathematicians who made important contributions to the field within the past three centuries adds further interest to the book.

The author has devoted much of the book to those special functions that emerge as solutions to the prototype differential equations of Physics. These functions are presented at a more generalized and rigorous mathematical setting than in many Mathematical Physics books aimed at beginning graduate students while sparing the more tedious proofs all too common in books on Functional Analysis, for example. In particular, his exposition of Green functions and Operator Theory are much more comprehensive and easy to follow than comparable treatment in other texts targeting the same readership.

Hassani continues on through Lie Groups to end this tome with a look at symmetries and conservation laws. The latter are especially relevant in Quantum Field Theory and HEP. Topics given cursory treatment in texts on these subfields of Physics are presented by Hassani in greater detail, again sparing the reader the mathematics that often serve more to sidetrack that edify.

Given the subject matter, it is no easy task to produce such a user-friendly tome, but Hassani has admirably risen to the task. I look forward to more texts by the author, perhaps one with more emphasis on Measure Theoretic approaches?

This is the best book that I have encounter so far, that functions as a sort of a "biblical" summary of almost all the basic (and no so basic) mathematics that is used in physics, and that also presents the material in a friendly accessible manner. I have gone through almost all the material. I must say that this book because of the features already mentioned and the fact that for each chapter has a well amount of problems to work out, becomes and indispensable tool of learning the basic (and no so basic) of mathematical physics. I also want to distinguish three more features, first, all the important concepts that are being treated have a note at the far left margin indicating the name of the theorem or name of the definition, this makes it easier to locate theorems and definitions. Second, there are plenty of examples that I have gone through and they are very instructive and third, the book is plagued with biographical notes about famous mathematicians and physicists which eases the atmosphere and are quite interesting.

The book is divided into thirty one chapters. Chapter 0 named "Mathematical Preliminaries" gives the basic ingredients that a course on real analysis has: sets, maps, metric spaces etc. Chapter 1 "Vectors and Transformations" deals with linear algebra, that is, vector spaces, linear transformations and algebras. Chapter 2 "Operator Algebra" deals with the basic operations that can be done with linear operators, Chapter 3 "Matrices: Operator Representations" is clear by its name what actually covers, Chapter 4 "Spectral Decomposition" deals with direct sums, invariant subspaces, eigenvalues and eigen vectors, spectral decomposition, functions of operators so that you can see that it basically touches upon the topics relevant to functional analysis only that a finite (and discrete) dimensional space is being used. Chapter 5 deals with Hilbert spaces, which are defined as infinite dimensional vector spaces which have an internal product and that are complete (this is, every Cauchy sequence of vectors in them converges to a vector contained within them), Chapter 6 deals with Generalized functions and this means essentially becoming acquainted with Dirac's delta function, Chapter 7 deals with Classical Orthogonal Polynomials, the approach used in this chapter is to derive all orthogonal polynomials from a single master formula where the difference enters in the parameters that this equation takes, thus presenting the Hermite, Laguerre, Legendre, Gegenbauer and Chebyshev polinomials. Chapter 8 deals with Fourier Analysis, this is, with Fourier series and Fourier transform, Chapter 9 with the basics of complex analysis going from analytic functions to Taylor and Laurent series, Chapter 10 with Calculus of Residues and Chapter 11 is called: Complex Analysis, Advanced Topics, here you see things like the concept of a meromorphic function, that is, a function whose poles are purely simple and have measured zero, also the MitagLeffler expansion of such a function is presented. Then the topics of Multivalued functions and Analytic continuantion are pursued and finally the Gamma and Beta functions are presented and the method of Steepest Descent. Chapters 12, 13, 14 and 15 deal with standard material of differential equations, in these chapters the author covers solutions to First order Differential Equations (FOLDEs), Second Order differential equations (SOLDEs), separation of variables in order to solve Partial Differential Equations (PDEs), in particular Chapter 14: Complex Analysis of SOLDEs is a delight. It deals with something that I always had judged as a sort of a mystery, why the series solution to an ODE may have an extra factor of the variable being summed, the so called Frobenius technique that results in solving a cuadratic equation called the indicial polynomial whose roots, the characteristic exponents, are the value sought after, of the power of the extra factor mentioned above. After this the way is pave to Fuchsian Differential Equations which in turn leads to the Hypergeometric and confluent hypergeometric functions. Chapter 15 deals with the technique of solving differential equations through the use of Integral transforms. Chapter 16 gives a condensed and easy introduction to point set topology and functional analysis, defining things like compact sets, compact operators, spectrum of compact operators and the spectral theorem for compact operators. Chapter 17 is named integral equations, and chapters 18 and 19 deal with Sturm-liouville systems. Chapters 20, 21, 22 deal with Green Functions (I didn't go properly through this chapters). Chapter 23 is about Group Theory a very important mathematical topic in Physics, I say that it is a good and very thorough introduction to the subject, the same happens with Chapter 24: "Group Representation Theory", but I recommend that you combine these chapters with other references and here I would like to recommend the book: Group Theory, a Physicist's survey of Pierre Ramond. Chapters 25 Algebra of Tensors introduces tensors as multilinear mappings, then goes to define the symmetric and exterior algebras ending in the Hodge star operator. Chapter 26 Analysis of Tensors, builds upon the previous chapter by applying the concepts of differential calculus to Tensors, in this way differential manifolds are defined, curves and tangent vectors, the very important concept of a differential of a map and with this base Tensor fields on manifolds are defined, then also Exterior calculus and Symplectic Geometry, the last one having a very important connection with Hamiltonian mechanics. Chapter 27 is my favorite, it deals with Lie Groups and Lie Algebras, it is the most transparent and general introduction to the subject I have encounter so far. Here we are told that a Lie group is basically both a Manifold and a group at the same time and the elements of the Algebra belong to the set of tangent vectors of the Lie group (manifold). Then Chapter 28 is called Differential geometry, what I like the most about this chapter is that before presenting the Curvature Tensor, it introduces the concept of the curvature two-form. The last 2 chapters I have not read them so I'm only going to cite their names: Chapter 29: "Lie groups and differential equations" and Chapter 30: Calculus of variations, symmetries and conservation laws.

Having read almost all of this book I have fell in love with it and I make a sharp guess that you will also. All in all, an excellent text to learn the basic (and no so basic) mathematics a Physicist needs and I only hope and wait for the author to write a second volume to cover higher mathematics (like algebraic topology for example) in the same easy-friendly style specially suited for Physicists. Thanks Dr. Hassani!

This is by far the best written book on the subject that I've seen. It is very clear and comprehensible. Modern and up-to-date, it is destined to become the standard book in its field.

Has anyone looked at the difference between Hassani's and C. D. Cantrell's book (Modern Mathematical Methods for Physicists and Engineers)? They seem to cover the same topics.