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I am already planning to attempt to learn programming in C and also mathematical proofs in some way since I've never had to attempt them before (I have a few books I can use, just not sure if I want to just go for really basic math concept proofs or go straight for the intro to real analysis books).

My question is: are there any other topics I should look into over the summer? I'm a math and physics major, and I plan to eventually go to graduate school for physics. I was thinking maybe partial differential equations would be a good thing to add, but wasn't sure (if anyone could recommend a book for that, it'd be great).