A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry 1st Edition by Peter Szekeres (PDF)

Description

Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students’ understanding of the various concepts, as well as to extend the text’s themes.

User’s Reviews

Editorial Reviews: Review ‘This is a beautifully crafted book. … Peter Szekeres presents in the most elegant and compelling manner a magnificent overview of how classic areas such as algebra, topology, vector spaces and differential geometry form a consistent and unified language that has enabled us to develop a description of the physical world reaching a truly profound level of comprehension. … Szekeres’s style is clear, thorough and immensely readable. His selection of topics concentrates on areas where a fully developed rigorous mathematical exposition is possible. … One cannot help but be slightly awed by the beauty and the capability with which seemingly abstract concepts, often developed in the realms of pure mathematics, turn out to be applicable … I recommend that you get hold of this book for yourself or for your library.’ The Times Higher Education Supplement’The superb layout and an index contribute to the excellent overall impression of this book …’. Zentralblatt MATH’ … the book may serve as an easily accessible introductory text on a wide range of the standard and more basic topics in mathematics and mathematical physics for the beginner, with an emphasis on differential geometry. a nice feature is that a considerable number of examples and exercises is provided, together with numerous suggestions for further reading: there is also an extensive index which will be particularly helpful for beginners in the subject.’ General Relativity and Gravitation Journal Book Description This textbook, first published in 2004, provides an introduction to the major mathematical structures used in physics today. About the Author Peter Szekeres received his PhD from King’s College London in 1964, in the area of general relativity. He subsequently held research and teaching positions at Cornell University, King’s College and the University of Adelaide, where he stayed from 1971 till his recent retirement. Currently he is a Visiting Research Fellow at that institution. He is well known internationally for his research in general relativity and cosmology, and has a good reputation for his teaching and lecturing. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book does not assume prior knowledge of the topics covered. However, the reader will find use of prior knowledge in algebra, in particular group theory, and topology. Compared to texts, such as Arfken Weber, Mathematical Methods for Physics, A Course in Modern Mathematical Physics is different, and emphasis is on proof and theory. The text is reasonably rigorous and build around stating theorems, giving the proofs and lemmas with occasional examples. The style is not the strictest, although making the text more reader friendly, it is easy to get confused with which assumptions have been made, and the direction of the proof. Sometimes only the “if” part is proven.Students familiar with algebra will notice that the emphasis is on group theory, interestingly the concept of ideals is left mostly untouched. For more on representation theory a good reference is Groups Representations and Physics by H.F. Jones where solutions to some of the exercises can be found, and examples of the use of the fundamental orthogonality theorem applied to characters of represenations.The first 6 chapters are relatively straight forward, but in chapter 7 Tensors the text becomes much more advanced and difficult. Chapter 10 on topology offers some lighter material but the reader should be careful, these consepts are to re-appear in the discussion of differential geometry, differentiable forms, integration on manifolds and curvature. These are not the most simple subjects and it is clear that they deserve entire courses of their own.The book has insight and makes many good remarks. However, chapter 15 on Differential Geometry is perhaps too brief considering the importance of understanding this material, which is applied in the chapters thereinafter. The book is suitable for second to third year student in theoretical physics.

⭐This book makes mathematical physics seem elementary. The author includes the required elementary undergraduate mathematics in early chapters, smoothly blending this into the mathematical physics in later chapters.Unlike some other attempts to create unified, integrated “mathematical methods for physics” books, this book is truly integrated. The earlier chapters are a gentle introduction to various pure mathematical topics which are required for mathematical physics, such as basic set theory and group theory, linear spaces and operators, and Lie algebras and Lie groups. Until this point, one could imagine that this is a pure mathematics textbook, particularly considering the choice of bibliography at the ends of chapters.Then there are two chapters on tensor algebra, which start to depart from the pure mathematical idiom and enter the physics-oriented way of thinking, with lots of indices which are, unfortunately, required for practical computations. This material leads smoothly into a chapter on physics (special relativity and relativistic electromagnetism).Then the focus returns to pure mathematics (topological spaces, topological vector spaces, measure and integration, Schwartz distributions, Fourier transforms, Green’s functions and Hilbert spaces). This material leads smoothly into another chapter on physics (quantum mechanics).Then there are about 125 pages (Chapters 15–18) on differential geometry, including some applications to thermodynamics and classical mechanics (Lagrangian and Hamiltonian mechanics) along the way. The differential geometry chapters seem to be devoid of the frustrating esoteric mysteries which are found in most DG introductions. The author makes a successful effort to demystify this subject area. These DG chapters lead smoothly into general relativity, the Schwarzschild solution (i.e. “black holes”), and some basic cosmological models.Finally there is a short chapter on Lie groups and Lie algebras.The main strengths of this book are the smooth blending of pure mathematics with mathematical physics, the demystification of many frustrating esoteric mysteries which are found in typical mathematical physics books, and the broad scope. This is an ideal gentle first introduction which allows the reader to actually understand what they are reading. After this point, mathematical physics does become totally incomprehensible and painfully esoteric. For a mathematical physics student, reading this book might be the last time they actually understand what they’re reading.

⭐For the intended audience (advanced students in theoretical physics) this is by far the best book available on the material it covers. The text is clear, the topics covered are presented in a logical sequence, and the student who works through the book will acquire a good background for understanding more advanced texts. Of course, not every area of mathematics used in theoretical physics is included; that would be impossible in a single book; but it is usable as a reference. In my opinion, however, this is mainly a book to study and work through, rather than a reference book. Exercises in the text help readers to check and solidify their understanding of the material.The author has obviously taken great care in the preparation of the book. There are very, very few typographical errors. Sadly, it is rare nowadays to find a book which has been as carefully proofread as this one.If you are a physicist and need to come up to speed on any of the topics covered by this book (one of the other reviews has helpfully listed the table of contents), waste no more time searching, just buy it.

⭐Since I don’t yet have this book, I cannot review it; however, I have found the contents of this book on the publisher’s web site in case it would help anyone decide to purchase it or not.ContentsPreface1. Sets and structures2. Groups3. Vector spaces4. Linear operators and matrices5. Inner product spaces6. Algebras7. Tensors8. Exterior algebra9. Special relativity10. Topology11. Measure theory and integration12. Distributions13. Hilbert space14. Quantum theory15. Differential geometry16. Differentiable forms17. Integration on manifolds18. Connections and curvature19. Lie groups and lie algebrasI will return at a later date to properly review it in case I need to change the rating I gave it.

⭐Caveat: I only started reading this a few days ago. So, not having finished it (or even close) please feel free to ignore this ‘review’, especially the star rating above.Still, I’m finding I like this book enormously. The explanations are clear and crisp, and the examples are helpful (though maybe not enough). And it’s certainly much better than the book I was looking at previously – Mathematics of Classical and Quantum Physics by Byron and Fuller. Also, though I haven’t looked at everything yet (or anything like), I’m impressed at how much the book aims to cover (the pace is a little brisk therefore, but that’s all right too as it keeps one from getting bored) – just the sort comprehensive coverage at an accessible level I was looking for. Finally, as someone with a maths rather than physics background, I much prefer the author’s decidedly ‘mathematical’ or ‘pure’ style of exposition to what you find in physics books (he makes a point of highlighting this at the outset, saying this a book on *mathematical* physics, a branch of mathematics, and not on theoretical physics, which is part of physics).But to tell the truth: I only started writing this in order to complain. *Why is this book so flipping prohibitively expensive?* At the moment I’m reading a PDF version downloaded from the internet on my Kindle. I was impressed enough to decide to buy this on here. But ah, the price! Fifty smackeroonies!! Oh dear oh dear! And not even a second-hand marketplace copy to be had. (I’m not a cheapskate; I’m just poor.)

⭐Dense. For physical intuition, this book expects you to work outside the box or have obtained it elsewhere. It is there. The exercises are genuinely challenging and it feels like each has a purpose. If you are looking for a book to focus hard on and are interested in the hessboard of mathematical physics – this is for you.

⭐Covers a lot of undergraduate ground. useful as a cross reference rather than main text book.

⭐表題はからは、数理中心で記述している「現代理論物理学」書と期待するが、中味は「現代物理学のための数学」書と認識してもらえば良い。表題より明らかなように現代物理学で要求される主な数学分野をバランス良くカバーしている。またあまり理論に凝らず基本的かつ実用的な程度で易しく書かれていて、英文も読みやすい。種々の該当分野の専門書を読む前に一読を薦める。また、ハンドブックとしても使える。物理に使う数学をひと通り書いただけの本です。群論，ヒルベルト空間論，微分幾何を一冊にまとめた教科書です。本書ならではの記述といったようなものは見受けられず，それぞれの分野の標準的な教科書を一冊にまとめた感じなので，改めて買う必要はないと思われます。

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