Recommended Math and Physics Books.
mostly grad level, but some undergrad.
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General undergrad advice: any books from Schaum's outline are great. Vector Analysis. Linear Algebra. Many worked problems. Kind of a Cliff's notes for math. Anything by Stillwell is great. Try, from time to time, to go to the source. For example, Einstein's ``The Meaning of Relativity' is orders of magnitude better than any other intro to special relativity that you can find.

General advice: first editions are almost always significantly better than subsequent editions. Authors seem compelled to add extraneous doo-dads and blather to fill up their original concise elegant volumes and turn its poetry into stinky prose. GET FIRST EDITIONS if you can. Anything by Milnor, Arnol'd, Atiyah, and Bott are worth reading.
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ODEs
Lectures on Ordinary Differential Equations, by W. Hurewicz . concise. beautifully written. rigorous. to-the-point
Ordinary Differential Equations, by Arnol'd.
`Differential Equations, Dynamical Systems, and Linear Algebra' by Hirsch and Smale, (*) 2nd best: 2nd ed. Hirsch, Smale, and Devaney

Mechanics (classical) Mathematical Methods in Classical Mechanics, by Arnol'd
vol. 1, Mechanics, Landau and Lifshitz
Foundations of Mechanics encyclopaedic
Celestial Mechanics, by Pollard
Symmetry in Mechanics, by Stephanie Singer. great undergrad level intro to reduction
Celestial Mechanics, Wintner.
Celestial Mechanics, Siegel-Moser.
notes by Chenciner
Goroff's introduction to his translation of Poincare's Les Nouvelles Methodes de Mecanique Celeste.

Intro Manifolds
An Introduction to Manifolds, by Loring Tu , with pdf here
An Introduction to Differentiable Manifolds and Riemannian Geometry (2nd ed) - William M. Boothby


Quantum Mechanics Principles of Quantum Mechanics, by Dirac.
Mathematical foundations of Quantum Mechanics, by George Mackey great for understanding the underlying mathematical structure


Bundles with Connections
one route in: start with:
``Vector Bundles with a Connection''' Chern, p. 1-27 in `Global Differential Geometry',MAA Studies in Mathematics, vol 27
move to:
``Lecture Notes on Elementary Topology and Geometry'' , chapter 7, Singer and Thorpe; phenomenal re Gauss-Bonnet on surfaces from perspective of said surfaces unit tangent bundle as a principal circle bundle.
follow up with :
Spivak, vol. 2. Find the commutative diagram that fills up a whole page. Decipher it.
then go wild:
Gauge Theory : Freed and Uhlenbeck; the amazing story of Donaldson's exotic R^4's
Gauge Theories , vol 5 of Atiyah's collected works. lecture;
Vortices and Monopoles, Jaffe and Taubes:
An encyclopaedic pre-Donaldson, not physics inspired reference:
Kobayashi-Nomizu, ch . 1 and 2. encyclopaedic. dry.

Riemannian Geometry and Analysis on Manifolds
Quick. To the point. Beautiful.
1. `Morse Theory' -Milnor, just ch. 7 ``A Rapid Course in Differential Geometry'
2. ``Mathematical Methods of Classical Mechanics'' -Arnol'd. just appendix 1. An excellent intuitive description of curvature

Detailed. Intense.
``Comparision Theorems in Riemannian Geometry''- Cheeger and Ebin:

Classics

With a physics bent; where I began ...
"Gravitation'' (a.k.a. `The Big Black Book')- Misner, Thorne, and Wheeler. good pictures. philosophy
`Foundations of Mechanics', - Abraham and Marsden.

Classic surface theory
H. Hopf. Springer vol. 1000. Beautiful. Beautiful problems.
Gray. `Volume of Tubes'. Why char. classes appear in looking at the expansion of the volume of a tube about a submanifold.


Differential and Alg. Topology
''Topology from the Differential Viewpoint'' - Milnor
`Morse Theory' -Milnor, again! (shoot, dang near anything by Milnor !)

`Introduction to Topology' -Vassiliev
Algebraic Topology, - Marvin Greenberg. get the1st edition, not the 2nd ! In the 2nd edition Marvin caught the familiar disease of trying to say everything he knows.

Homotopic Topology by Fuchs and Fomenko. added summer 2016: Fuchs is finishing up a polished edition. Worth buying! CW complexes and homology, done right.
Hatcher , the post 2010 standard
PDE Linear Differential Operators , by Cornelius Lanczos. (Alex Castro suggested, fall 2011!)
Lectures on Partial Differential Equations (ISBN: 3540404481) by V. I. Arnol'd. Wow! read what Maxwell did for spherical harmonics
Analysis: Simmons' Introduction to Topology and Modern Analysis This is an excellent book, at a level between the usual undergrad analysis, and grad analysis courses.
Measure theory: Royden;
Fractals, Hausdorff Measure (for us beginner's): Falconer.
Stokes' theorem, beginning calc. on mfds: Spivak's ``Calculus on Manifolds''
Algebra: Herstein
Algebra , Artin



UNDERGRAD .
Geometry: anything by Stillwell. anything by Coxeter. `Geometry and the Imagination', by Hilbert and Cohn-Vossen.

GENERAL ADVICE: Almost All Schaum's Outlines Schaum's Outlines on `Vector Analysis', `Linear Algebra', `Real Analysis' b
the Schaum style is a short telegraphic section on theory , 1 to 3 pages, followed by scores of worked problems. Each volume has 100s of worked problems. These do-it-yrself books provide a good, quick and dirty way to learn lots of math.
vector calculus, differential forms:
Schaum's Outline on Vector Analysis (
Div Grad Curl are Dead -- by W. Burke.

Linear algebra : Hoffmann and Kunze
Relativity: Einstein's The meaning of relativity is the best book for learning special relativity. *** generally speaking it is best to learn a subject from the person who invented it.


Alex Castro recommendations.
(I have not perused these in depth, but Alex has good taste.)

Mathematical Omnibus: Thirty Lectures on Classic Mathematics [Hardcover] Dmitry Fuchs (Author), Serge Tabachnikov (Author)
Modern Geometry - Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (Pt. 1)[Hardcover] B.A. Dubrovin (Author), A.T. Fomenko (Author), S.P. Novikov (Author), R.G. Burns (Translator)
Modern Geometric Structures And Fields (Graduate Studies in Mathematics)[Hardcover] S. P. Novikov; I. A. Taimanov (Author)
** new version of Modern Geometry vol. 1 -- Novikov has had some tension with Fomenko since the latter became a "historian", and is trying to get rid of any clues which affiliates them. The new book has cleaner notation, and shorter proofs.
Mathematical Analysis I and II (Universitext) [Paperback] V. A. Zorich (Author), R. Cooke (Translator)

Vinberg's "A Course in Algebra" ** I used it for my prelim
Gilbert Strang's "Calculus" (Free) and "Intro to Linear Algebra **awesome books
Tales of Mathematicians and Physicists [Paperback] Simon Gindikin (Author), Alan Shuchat (Translator)
** in the same spirit of Arnold's biography of Newton
Riemann, Topology, and Physics (Modern BirkhC$user Classics) [Paperback] Michael I. Monastyrsky (Author) ** a lot of fun -- bedtime stuff