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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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

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.

An Introduction to Manifolds, by Loring Tu , with pdf here

An Introduction to Differentiable Manifolds and Riemannian Geometry (2nd ed) - William M. Boothby

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

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:

An encyclopaedic pre-Donaldson, not physics inspired reference:

Kobayashi-Nomizu, ch . 1 and 2. encyclopaedic. dry.

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.

Hatcher , the post 2010 standard

Measure theory: Royden;

Fractals, Hausdorff Measure (for us beginner's): Falconer.

Algebra , Artin

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.

Schaum's Outline on Vector Analysis (

Div Grad Curl are Dead -- by W. Burke.

(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