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