I'm a 12th grade student and I've recently developed a passion for mathematics . Currently my knowledge in this particular area is comprised by : single-variable calculus , trigonometry , geometry , basic notions of linear algebra and set theory .

I'm particularly interested in calculus and I need some advice as I intend on coupling my future math skills with the study of Quantum Physics .

Where should I go from here in order to understand multivariable calculus ?Can anyone recommend an interesting advanced calculus textbook ?

(Hope the prerequisites needed for understanding the textbooks you suggest match my current mathematical skills and knowledge)

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    $\begingroup$ If you've studied basically high school calculus it is very likely you still need quite a way to cover in one single variable calculus. Special attention to theorems and their demonstration will be a wise move. Complete this with some more thorough linear algebra I, and you shall be more than ready to begin superior studies everywhere (or almost). You can enter most serious universities' Mathematics Departments sites and read there the syllabus of these courses. $\endgroup$ – DonAntonio May 4 '14 at 12:31

As DonAntonio commented a good idea is to review single variable calculus emphasizing the theorems and their proofs. A good book to go beyond just the praxis of solving integrals and into the demonstrations is Tom Apostol's "Calculus and Linear Algebra" volumes 1 and 2. The first is primarily concerned with single variable calculus and analytic geometry, the second volume with proper linear algebra and multivariable calculus, with nice add-ons like differential equations, probability and some numerical analysis. I do think both volumes address a "starters set" in both math and physics.

Regarding quantum mechanics, although linear algebra, differential equations and Fourier analysis are usually enough to grasp the methods I strongly suggest you invest in learning the classical physics first, as in classical mechanics, thermodynamics and electromagnetism. An option is to try Feynman's Lectures that are now fully available online in high resolution here http://www.feynmanlectures.caltech.edu/

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