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I will be attending Notre Dame this following fall, and I am hoping to major in physics, as well as research (I haven't really decided what I will to focus on, esp. at the graduate level, but I hope I can figure that out later on down the road).

Over the summer, I want to independently study math, just to be familiar with it and have a decent background in it (My physics teacher said that he had wished he had a stronger math background before he was taking certain classes, as opposed to studying math WHILE he was in the classes, because he thinks the two would have linked together and explained things a tad bit better.)

As far as my current level of math, I seem to catch on and understand concepts, formulas, etc. very well. On the other hand though, my school has disappointing math offering, so I have Algebra II + Trig, but each had a very narrow focus, and I don't have much of a background in series or trig identities, and maybe other topics as well. For calculus, I had some level of independent study starting in Sophomore year ( I wanted to find, analytically, how to find turning points of functions, and my teacher wouldn't tell me because I "just wouldn't be able to understand it", and I found calculus very interesting, etc.) Also, I took a Brief Calculus I and II at a community college, which, as far as I can tell, was a survey of basic calculus topics without any trig at all. I only took the class to get out of my AP class, b/c the calc teacher was a jerk and wouldn't let me take it without Probability and Statistics, and he taught too slowly anyways.

The calculus that I know (and trig/algebra, for that matter), though, I know VERY well, at the conceptual level. It all just makes sense to me.

As for Physics, I took a standard AP B class with the Giancoli book, and a wonderful, fabulous teacher. I really haven't struggled at all with it (except maybe with visualizing E&M, right at first, but I could always use the formulas just fine).

As far as I can tell, I need to study Calculus I through IV very thoroughly, as well as diff.eq, linear algebra, and complex analysis. I have books on the diff.eq, linear algebra, and complex analysis, but I am really wondering what order I should approach these (and which Calc book I should buy, given my background), as well as any other suggestions for books/topics I should look into. If it is worth anything, I am considering a focus in theoretical physics, and I am not (necessarily) expecting to get through everything this summer, but I am dedicating most of the summer to my studies.

I apologize for the length, as this post definitely grew to be much longer than I had intended it to, and I thank everybody for all help in advance!