I am searching for a book to study physics. So far, I've been suggested

Resnick, Halliday, Krane, Physics,

but it doesn't seem to be very suited for a math major. Can you suggest some more mathematical books?

By mathematical I mean:

  • rigorous, with complete good proofs;
  • at the same time, intuitive and throughout in explanations.
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    I would have a look at this bibliography: https://www.ocf.berkeley.edu/~abhishek/chicphys.htm – user179549 Sep 29 '14 at 06:11
  • Arguably the most important mathematical method in physics is microadditivity which is presented clearly in A Primer of Infinitesimal Analysis by John L Bell. Other books are more obscure about it. –  Sep 29 '14 at 07:22
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    Classical Mechanics by John Taylor, or Classical Mechanics by Goldstein, would be a good place to start. – Monk Sep 29 '14 at 08:52
  • Roger Penrose's stuff is presumably mathematical. – rlms Sep 29 '14 at 12:48
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    Do you want to learn physics or the mathematics behind it? Physics requires some phenomology intuition which cannot be explained by rigor with complete proofs. – Mark Fantini Oct 01 '14 at 20:37
  • @MarkFantini I would like to learn physics and understand the intuitive and phenomenological side of it, but I would like to study such concepts from a rigorous point of view. – Dal Oct 01 '14 at 20:46
  • That is in some respects incompatible. There are also many parts of physics that haven't been made rigorous yet. It also depends on where you draw the line for rigor. – Mark Fantini Oct 01 '14 at 20:47
  • I wonder if anybody has mentioned "General Relativity for Mathematicians" by Rainer Sachs H.-H. Wu, which really requires a strong mathematical background. – No One Oct 01 '16 at 03:01

8 Answers8


There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether or not that's true is debatable. In any event, here's a list to get you started:

Lectures on Quantum Mechanics for Mathematics Students by L. D. Faddeev and O. A. Yakubovskii: Really the only one for undergraduates-a beautiful and justly famous Russian treatment now in English. But even this beginning text requires a good knowledge of differential equations, real and complex analysis, linear algebra and group theory.

Quantum Mechanics for Mathematicians by Leon A. Takhtajan : One of several such texts for second year graduate students, most of which need at least first year graduate courses in analysis,differential geometry and topology. Excellently written,though.

Physics for Mathematicians, Mechanics I by Michael Spivak : Awesome book by the master. Perfect follow up to his 5 volume opus on graduate differential geometry and shows in depth what all that beautiful manifold theory was good for. Let's hope he finishes the projected other 3 volumes.

Quantum Theory for Mathematicians by Brian C. Hall A recent addition to the list I haven't seen. But if it's as good as his Lie groups book, it'll be well worth checking out.

Mathematical Methods in Quantum Mechanics by Gerald Teschl : This book is really more about the methods of quantum theory then the science itself. That being said,it's by a master and presents the material beautifully for second year graduate students.

Quantum Field Theory by Gerald B. Folland: Another terrific textbook by Folland, covering everything about QFT that can be made rigorous at this point,which sadly isn't as much as we'd like. You better have your graduate analysis chops on before tackling this one.

Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill : Exactly what the title says it is. A classic,one of the best books on relativity theory for mathematicians. The book to read after Spivak.

Modern Geometric Structures and Fields by S. P. Novikov and I. A. Taimanov: An incredible and completely unorthodox beginning graduate course in differential geometry by one of the premier mathematicians in the world that completely interweaves the careful theory with virtually the entire modern structure of physics-from basic mechanics through relativity theory through the beginnings of string theory and quantum field theory.A good working knowledge of rigorous calculus of one and several variables and linear algebra is all you need to tackle this one. An absolute must have if you're interested in the relationship between mathematics and physics. I DO wish it had more exercises.

UPDATE: I've added a few more in response to some of the other posts here:

Foundations of Mechanics 2nd edition by Ralph Abraham and Jerrold E. Marsden: A remarkable and careful text for both physicists and mathematicians giving one of the first detailed presentations of classical mechanics from the modern differential geometric point of view. Very advanced, really for strong graduate students in mathematics or very strong graduate students in physics. A good follow up to Spivak's mechanics book and a first year graduate mathematics course sequence.

Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems by Jerrold E. Marsden and Tudor S. RatiuL A very mathematical first year graduate course in classical mechanics,again from the moder geometric point of view and emphasizing symmetry. Can act as the preliminary text for the text immediately preceding this one. About the same level as Spivak,but it's not as rigorous and focuses more on advanced physics then geometry. Very good collateral reading to that text.

Symplectic Techniques in Physics by Victor Guillemin and Shlomo Sternberg: I've never seen this book, but I've heard good things about it. It's by 2 masters,so that by itself makes it worth checking out.

Ordinary Differential Equations by V.I. Arnold: A terrific intermediate level course emphasizing the role of linear transformations and manifolds in the study of linear ODEs. A strong course in linear algebra and geometry is needed; this beautiful text is a must for students interested in the geometry of differential equations and physics.

Geometrical Methods in the Theory of Ordinary Differential Equations by V.I. Arnold: Graduate level follow up to the preceding text. An introduction to ordinary differential equations and dynamical systems emphasizing thier role in chaos theory and physics. A very difficult and sometimes mystifying book, but the sheer richness of the ideas and the interplay of math and physics makes it worth the effort.

I'll also mention in passing at the end the wonderful physics textbooks by Walter Griener. Not only are they concise,wide ranging and extremely clear, they have more solved examples then any physics book I've ever seen. They're written by a master. If you can get them in the original German and can read those,that would be even better as some errors creeped in in translation.

Hope that gets you started. Good luck!

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    Wow, this is great, knew about none of this but Spivak! Bookmarking... – Kevin Arlin Sep 29 '14 at 07:14
  • Spivak's book is a good one to start with, I guess. Thank you very much for the valuable suggestions. – Dal Sep 29 '14 at 13:20
  • You should know, since it appears you've not read it, that Resnick et al. is a middle school level heuristic survey of random topics in physics used to write GRE questions and nothing else. Physics for poets, basically. One need not go so advanced as the things you've listed in order to get something "mathematical" compared to it. –  Sep 30 '14 at 04:41
  • @Chris I've glanced through Resnick,but no,mot fully familiar with it.I actually learned basic physics from Serway,another of the standard textbooks.What that has to do with my post,I'm somewhat perplexed about.......... – Mathemagician1234 Sep 30 '14 at 04:56
  • I'm just saying the OP may not know what they're asking for. Someone who's being taught out of that book is not in a position to understand *any* of the physics in the books you've mentioned, and probably very little of the math. –  Sep 30 '14 at 04:58
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    @Chris Uh,that was kind of the point of my opening paragraph,hello? As far as I know,there really aren't any basic physics textbooks of the level he's asking for.To be honest,I'm not sure how to design one that doesn't end up looking like a basic physics text with some proofs thrown in that would confuse most students at that level! – Mathemagician1234 Sep 30 '14 at 05:00
  • Dear @Mathemagician1234, I'm now starting studying Spivak's. I have read in the preface that knowledge of differential geometry is assumed, but is it really necessary to understand at least **part $1$ and $2$**? – Dal Dec 23 '14 at 00:35
  • @Mathemagician1234: also, by the way, would you mind telling me what is the require background to study differential geometry (for example, these notes I've just found: https://www.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf)? – Dal Dec 23 '14 at 00:39

Here is another bunch of texts. Like the ones suggested by Mathemagician1234, they are not general texts. The level of formality is variable.

Classical mechanics

F. Scheck, Mechanics, Springer, 2010. Although not specifically geared toward mathematicians, it makes use of mathematically advanced tools. I consider it the best book on classical mechanics currently available, much superior to Goldstein.

A. Fasano, S. Marmi, Analytical Mechanics, Oxford University Press, 2006. Thorough and complete textbook, strongly mathematically oriented (at undergraduate level).

N. M. J. Woodhouse, Introduction to analytical dynamics, Springer, 2009.

Quantum mechanics:

S. J. Gustafson and Israel, Mathematical concepts of quantum mechanics, Springer, 2011.

F. A. Berezin and M. A. Shubin, The Schroedinger equation, Kluwer, 1991.

F. A. Berezin, The method of second quantization, Academic Press, 1966.

Special relativity

G. L. Naber, The geometry of Minkowski spacetime, Springer, 2010. This is superbly written. There are also two other books from the author on gauge theories.

N. M. J. Woodhouse, Special relativity, Springer, 2003.

Statistical mechanics

F. A. Berezin, Lectures on statistical physics, Online

R. A. Minlos, Introduction to Mathematical Statistical Physics, American Mathematical Society, 2000.

Last but not the least I'd like to suggest two (wonderful) calculus books which, instead, are full of examples taken from physics:

V. A. Zorich, Mathematical Analysis I and II, Springer, 2004.

Massimo Ortolano
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    +1 for a very good list,especially Zorich. Zorich's main flaw, sadly,is it lacks exercises. It'll really need to be supplemented with exercises to be effective as a textbook even for honors students and it would be quite hard to find appropriate ones. – Mathemagician1234 Sep 30 '14 at 03:08

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics".

As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult to get a good understanding of the later without insight into Lagrangian and Hamiltonian Mechanics.

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At Oxford we used http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710 for the first year of the Physics Course.

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I really wondering why nobody has already mentioned this classic books:

Classical Mathematical Physics - W. Thirring

Quantum Mathematical Physics - W. Thirring

If someone already knows a lot about Topology and Differential Geometry, Chapters 1, 12 and 13 from

Geometry, Topology and Physics - M. Nakahara

will give some nice introduction to the applications of modern math to Physics

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Borisenko-Tarapov, Vector and Tensor Analysis with Applications, Dover Books. I liked it when I was young and just loaded it down as pdf. But perhaps this more is very classical physics, but not so general mathematics book..?

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Try Principles of Physics by Resnick, Halliday and Walker

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    Horrible book, -1. About as far as one can get from a mathematical treatment of physics. – zzz Jun 16 '15 at 22:49