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If you're really dedicated to learning the mathematics necessary for advanced physics, I know of a certain pure math book that is fairly remarkable in its depth and breadth of coverage.
It's called Advanced Calculus and is by Sternberg and Loomis. You can find a link to it for free in PDF form http://www.math.harvard.edu/~shlomo/" .
It assumes you know all the traditional material contained in a standard calculus sequence. It also assumes a truly unusual amount of mathematical maturity for students at the freshman level.
You must be very dedicated and willing to work hard to get through it, but it will take you through abstract linear algebra, rigorous single-variable and multivariable calculus, the foundations of the theory of integrals, and finally on to the calculus of manifolds using tensor analysis and multilinear algebra. It essentially contains an entire undergraduate education in analysis (and probably a bit of a graduate-level education in analysis as well at all but the most accelerated schools). It also has a section somewhat in the middle on differential equations.
Have fun!
As a side note, in my opinion this book is a prime example of how poor mathematics education is in the U.S. I find it sad that many people graduating with undergraduate degrees in mathematics in the U.S. would probably find much material in that book they don't know, even though it was intended to be used as a two- or three-semester analysis sequence for entering freshmen.
It's called Advanced Calculus and is by Sternberg and Loomis. You can find a link to it for free in PDF form http://www.math.harvard.edu/~shlomo/" .
It assumes you know all the traditional material contained in a standard calculus sequence. It also assumes a truly unusual amount of mathematical maturity for students at the freshman level.
You must be very dedicated and willing to work hard to get through it, but it will take you through abstract linear algebra, rigorous single-variable and multivariable calculus, the foundations of the theory of integrals, and finally on to the calculus of manifolds using tensor analysis and multilinear algebra. It essentially contains an entire undergraduate education in analysis (and probably a bit of a graduate-level education in analysis as well at all but the most accelerated schools). It also has a section somewhat in the middle on differential equations.
Have fun!
As a side note, in my opinion this book is a prime example of how poor mathematics education is in the U.S. I find it sad that many people graduating with undergraduate degrees in mathematics in the U.S. would probably find much material in that book they don't know, even though it was intended to be used as a two- or three-semester analysis sequence for entering freshmen.
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