Recommended Math and Physics Books.

Added, June 2012: ``From Calculus to Cohomology'' by Madsen and Tornehave, Princeton U. Press.
This book is perhaps the best upper division intro to forms and topology. Sits in between the levels of Spivak's `Calculus on Manifolds' and Bott and Tu. Recommended by Martin Weissman.

Riemannian Geometry, Analysis on Manifolds:
ch 7 of ``Morse Theory'' , by Milnor, chapter title: ``A Rapid Course in Differential Geometry'; the best brief intro to Riemannian geometry

Appendix 1 from Arnol'ds ``Mathematical Methods of Classical Mechanics'' excellent brief intuitive description of curvature

Riemannian Geometry, by Jack Lee. good intro., choice of topics, not too wordy.
Do Carmo's ``Riemannian Geometry'' -- becoming a standard.
``Riemannian Geometry'' -- Willmore; good place for the discussion of the algebra of the space of curvature tensors
Kobayashi-Nomizu, vol. 1, esp. ch 1 and 2 for principal bundles and connections thereon;
Spivak, vol. 2
(Why don't I have Spivak's 5 volume set ? He does not know how to edit himself. But if you've got LOTS OF TIME, it's fun. Esp. the translation of Riemann in vol 1.)
`` Gravitation'' by Misner, Thorne, Wheeler et al. good pictures and philosophy
`Foundations of Mechanics', by Abraham and Marsden, all the formulas a geometric mechanician might ever need.

Topology
Milnor's ''Topology from the Differential Viewpoint''
Milnor's Morse Theory.
shoot, pretty much any book by Milnor !
Vassiliev's `Introduction to Topology'
Algebraic Topology, by Marvin Greenberg. 1st edition is better. Tries to say too much in 2nd.
Homotopic Topology by Fuchs and Fomenko. Hard to get in English! horrible mess of an index, but worth the pain. CW complexes and homology, done right.

PDE Linear Differential Operators , by Cornelius Lanczos. (Alex Castro suggested, fall 2011!)
Partial Differential Equations by V. I. Arnol'd.
Analysis: Simmons' Introduction to Topology and Modern Analysis This is an excellent book, in between the usual undergrad analysis, and grad analysis courses.

Stokes' theorem, beginning calc. on mfds: Spivak's ``Calculus on Manifolds'' Measure theory: Royden;
Fractals, Hausdorff Measure (for us beginner's): Falconer.
Algebra: Herstein
Algebra , Artin

Gauge Theory : Freed and Uhlenbeck;
Atiyah's lecture;
Jaffe and Taubes: Vortices and Monopoles.
Mechanics: Arnol'd's Mathematical Methods in Classical Mechanics'
Landau and Lifshitz
Abraham and Marsden `Foundations of Mechanics', encyclopaedic; start above, use this as a ref.
Celestial Mechanics
.
Wintner. Siegel-Moser. notes by Chenciner
Goroff's intro to his translation of Poincare's Les Nouvelles Methodes de Mecanique Celeste.

Quantum Mechanics Dirac's Principles of Quantum Mechanics
George Mackey's mathematical foundations of Quantum Mechanics is excellent for understanding the mathematical structure


UNDERGRAD .
Ordinary Differential Equations 1. Hirsch and Smale `Differential Equations, Dynamical Systems, and Linear Algebra' (preferred; but out-of-print). Or Hirsch, Smale, and Devaney `Differential Equaions, Dynamical Systems, and ..' (expanded version of Hirsch-SSmale) w some `chaos' added. Arnol'd. `Ordinary Differential Equations'. Harder.
Geometry: anything by Stillwell. anything by Coxeter. `Geometry and the Imagination', by Hilbert and Cohn-Vossen.


Celestial Mechanics : Pollard's `Celestial Mechanics'.
Stephanie Singer's `Symmetry in Mechanics'. 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