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Favorite math books for physics self-study

gengogakusha "gengogakusha"
The list author says: "Some books have tags:[UG] undergraduate level; [AUG] advanced UG; [G] graduate level.

Most of the math and physics I know I've taught myself. My experience has been that it's generally a mistake to try to learn mathematics from physics books.

Picks are biased toward relativity, quantum mechanics and elementary particle physics.

More specialized or advanced material is toward the beginning. Search down until you reach your comfort zone :)

Hopefully my experience as reflected in this list will be of help to you!"
Matrix Groups for Undergraduates (Student Mathematical Library,)
Matrix Groups for Undergraduates (Student Mathematical Library,)
"[AUG] Phenomenally good intro to Lie groups and Lie algebras via matrix groups. Very concise: about 137 pp ex exercises (no solutions). Prereqs: multivariable calculus & basic analysis; abstract algebra (groups, fields, morphisms, conjugation) and linear algebra. Fast pace, no frills but crystal clear.  Must be the best intro by a long shot! Author's AMS webpage has 10th ch on roots and errata."
Groups and Symmetries: From Finite Groups to Lie Groups (Universitext)
Groups and Symmetries: From Finite Groups to Lie Groups (Universitext)
"[AUG] Extremely concise but well-written. For math majors so be prepared - must be comfortable with abstract algebra among other areas!  If you can hack the math, it's a great introduction to math powering the Standard Model of elementary particles!"
Tensor Analysis With Applications
Tensor Analysis With Applications
"[AUG] Concise, modern introduction geared toward relativity!  Having struggled with tensors and been frustrated by the inadequate exposition in relativity books, I was happy to discover this beauty! Check it out via the Search Inside function.  Great price too!  Cf. Kay's "Schaum's Outline of Tensor Calculus" below."
Schaum's Outline of Tensor Calculus (Schaum's)
Schaum's Outline of Tensor Calculus (Schaum's)
"[AUG] Solid intro for self-study based on older components approach; covers many topics incl. special relativity. For special and general relativity, learning tensor calculus is a must and this is one of the best - and by far cheapest - book I've found on the subject.  My first pick though is Ahsan's "Tensor Analysis with Applications", especially if you're into relativity."
Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics)
Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics)
"[G] If you want the real math underlying general relativity, here it is, in a pedagogically brilliant form.  Really for mathematicians so be forewarned strong math skills including previous background in differential manifolds required"
Tensors and Manifolds: With Applications to Physics
Tensors and Manifolds: With Applications to Physics
"[G] Lucid mathematical development, written in a very readable style.  Must know advanced linear algebra (e.g., operators), some abstract algebra e.g. modules), typical classical math for grad physics or math majors.  Unique, I think, and really superb."
Introduction to Spectral Theory in Hilbert Space (Dover Books on Mathematics)
Introduction to Spectral Theory in Hilbert Space (Dover Books on Mathematics)
"[G] Superb intro to HS theory! Clear, careful, concise exposition for those not wanting to fill in important material via exercises, a practice I detest.  Prereqs: linear algebra; some analysis, topology, group theory. Nice pace: Hilbert spaces on p. 23; spectrum p.158. Has spectral analysis of unbounded self-adjoint operators. Anyone interested in HS formalism of QM should check it out!"
An Introduction to Manifolds (Universitext)
An Introduction to Manifolds (Universitext)
"[AUG/G] Lucid presentation, concise & self-study friendly.  Covers fewer topics than Lee's Intro to Smooth Manifolds and is less detailed. So you won't get "lost in the trees". I used both books to great advantage. Tu is one of my favorites!  Make sure you get the new edition (pictured)!"
Introduction to Smooth Manifolds (Graduate Texts in Mathematics)
Introduction to Smooth Manifolds (Graduate Texts in Mathematics)
"[G] Amazing book, best in class for self-study. Anyone wanting to understand the mathematics underlying general relativity would profit from reading this book.  It is long but that's because Lee provides a lot of clarifying discussion along with the math. But make sure you have the requisite background first."
Elementary Differential Geometry, Revised 2nd Edition, Second Edition
Elementary Differential Geometry, Revised 2nd Edition, Second Edition
"[AUG] Excellent for self-study. Very notation heavy topic so requires perserverance! Many people also recommend Pressley, Elementary Diff. Geo, but overall I preferred this one. Great preparation for more advanced book like Lee, Intro to Smooth Manifolds, or Tu's An Introduction to Manifolds."
Elementary Differential Geometry
Elementary Differential Geometry
"[AUG] Haven't studied this new [2010] contender but from what I've seen via google books or Amazon's search inside function, this looks like a really nice, concise introduction suitable for self-study.  Recommend checking it out before deciding on a first book."
Linear Functional Analysis (Springer Undergraduate Mathematics Series)
Linear Functional Analysis (Springer Undergraduate Mathematics Series)
"[AUG/G] Provides necessary background for understanding the math underlying quantum mechanics.  Exceptionally well-written, suitable for self-study.  I really enjoyed the presentation."
Introductory Functional Analysis with Applications
Introductory Functional Analysis with Applications
"[AUG] Deservedly a classic! Fine for self-study; covers a lot of territory and has answers to odd questions. Recommend first getting some background in topology and metric spaces, not to mention linear algebra and calculus."
Algebra (AMS Chelsea Publishing)
Algebra (AMS Chelsea Publishing)
"[AUG/G] 3rd edition (1988). A classic and for a good reason! Although it assumes a great deal of mathematical maturity (i.e., ability to deal with a high degree of abstraction), it is very clearly written. I have no idea what more recent abstract algebra books are like but this classic is quite enjoyable to read (well, I haven't quite read it all yet :)."
Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)
Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)
"[G] This classic used to be way over my head but over the years, I've really come to appreciate it.  Amazingly concise and lucid exposition; no fluff, no frills, just math exposition at its very best. For the math mature so read an introductory book first."
Introduction to Analysis (Dover Books on Mathematics)
Introduction to Analysis (Dover Books on Mathematics)
"[AUG] Nice first book on real analysis good for self-study.  Concise (about 250 pp). Covers set theory basics; real numbers; metric spaces; continuous functions; differentiation; Riemann integration; partial differentiation; multiple integrals. No answers to problems and no Lebesgue integration. Cf. the Amazon "search inside function".  A high quality Dover bargain!"
Advanced Linear Algebra (Graduate Texts in Mathematics, Vol. 135)
Advanced Linear Algebra (Graduate Texts in Mathematics, Vol. 135)
"[G] Exceptionally well-written, concise reference text. Mathematical poetry but definitely def., theorem, proof style. Must be comfortable with both basic linear algebra and abstract algebra (e.g. groups, rings) and that level of abstractness to productively use this book. But if you've paid your math dues, this book is tops.  If others confuse me or I don't know something, I look here."
Linear Algebra Done Right (Undergraduate Texts in Mathematics)
Linear Algebra Done Right (Undergraduate Texts in Mathematics)
"[AUG] I loved this book: great for making the transition from undergraduate linear algebra to the real stuff. Only drawback is there are no solutions to exercises, which is irritating.  For details, please see my Amazon review."
Applied Linear Algebra
Applied Linear Algebra
"[AUG] Great for self-study, especially if you want some help moving from basic UG to more advanced linear algebra and are also particularly interested in applications to physics.  Check it out!"
Metric Spaces (Springer Undergraduate Mathematics Series)
Metric Spaces (Springer Undergraduate Mathematics Series)
"[AUG] Excellent and thorough introduction, very clear; highly recommended for self-study.  Includes solutions to exercises and a handy appendix reviewing necessary mathematical concepts and definitions. For more details, see my Amazon review."
Topology (2nd Edition)
Topology (2nd Edition)
"[AUG] Very clearly written book.  Has more than what's needed for a book like Lee, Intro to Smooth Manifolds or other intro differential geometry books. For a refresher, I consult Munkres. But for 1st exposure I recommend Mendelson, which goes from metric spaces to topological spaces. I have the 1st ed (1977)."
General Topology (Dover Books on Mathematics)
General Topology (Dover Books on Mathematics)
"[AUG/G] Very nice topology book for the mathematically more mature at a very nice price. Includes general convergence via theory of nets and theory of filters.  Note this book first published in 1970 and so reflects the terminology and style of that time."
Introduction to Topology: Third Edition (Dover Books on Mathematics)
Introduction to Topology: Third Edition (Dover Books on Mathematics)
"[AUG] Admirably clear, well organized, concise yet has sufficient examples.  Covers basics of many key topics but some core topics like separation or metrization theorems aren't in it. Moves from more intuitive metric spaces to abstract topological spaces in a nice way. For 1st exposure I like the organization better than that of Munkres. Reread it for this review and still love it."
Ordinary Differential Equations (Dover Books on Mathematics)
Ordinary Differential Equations (Dover Books on Mathematics)
"[AUG/G] Tops for self-study but only for the mathematically mature who do not mind an older-style book (and small font). If you're not confident in your math skills, likely not the best choice for 1st exposure. Chock full of fully worked examples. Has short Lessons with good cross-references so they can be read out of sequence. A classic; great buy!"
Differential Equations
Differential Equations
"[AUG] I found Polking, Boggess & Arnold (1st ed., 2001, ISBN 0-13-598137-9) well-organized, clear and very "well exampled", pace the negative reviews.  Not sure why some hate it: my guess - they're simply not ready for "Diffy Q". Requires less math maturity than Tenenbaum. Nice graphics. Fine for self-study. Forget those whimpy whinners and jump on in! 2nd ed might be better but I don't know."
Linear Algebra: A Modern Introduction- Text Only
Linear Algebra: A Modern Introduction- Text Only
"[UG] Best introduction I've found. I have the 2nd ed (shown).  There's a newer, more expensive one that could be better.  Overall, I liked this book better than Lay's, but Lay is also good."
Multivariable Calculus: Concepts and Contexts (Available 2010 Titles Enhanced Web Assign)
Multivariable Calculus: Concepts and Contexts (Available 2010 Titles Enhanced Web Assign)
"[UG/AUG] Superb book: clear exposition, beautiful diagrams. This is where I look when I forget something about vector calculus. Note that this is a practical (engineering-oriented) book, for theoretical treatments see a book on analysis such as Rosenlicht or Rudin. I have the 1st ed. (1998)."