Best books for mathematical background?

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What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?

Some subjects off the top of my head that probably need covering:

• Differential geometry, Manifolds, etc.
• Lie groups, Lie algebras and their representation theory.
• Algebraic topology.

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user ahh

asked Nov 4, 2010
recategorized May 4, 2014

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The last book I read on "background in math for physicists" was "Mathematics for Physics" by Stone and Goldbart, and I enjoyed it quite a bit. (Since then I've tended to hit the pure math books, but that's a different story).

Even better, a version of the book is available online at Paul Goldbart's webpage.

Here's a list of topics:

* Calculus of Variations
* Function Spaces
* Linear Ordinary Differential Equations
* Linear Differential Operators
* Green Functions
* Partial Differential Equations
* The Mathematics of Real Waves
* Special Functions
* Integral Equations
* Vectors and Tensors
* Differential Calculus on Manifolds
* Integration on Manifolds
* An Introduction to Differential Topology
* Groups and Group Representations
* Lie Groups
* The Geometry of Fibre Bundles
* Complex Analysis I
* Complex Analysis II
* Special Functions and Complex Variables
o Appendix A: Linear Algebra Review
o Appendix B: Fourier Series and Integrals

This post imported from StackExchange Physics at 2014-05-04 07:44 (UCT), posted by SE-user j.c.
answered Nov 11, 2010 by (260 points)
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Sean Carroll's Lecture Notes on General Relativity contain a superb introduction to the mathematics of GR (differential geometry on Riemann manifolds). These also also published in modified form in his book, Spacetime and Geometry.

Spivak's Calculus on Manifolds is a gem.