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It seems like physicists have a working knowledge of mathematics rather than a really deep knowledge of it. I realized this after trying to show all of my work on a simple fraction simplification problem. I didn't know how. I know how to get the right answer, but I didn't actually know how to show the formalisms to my work, and explain the steps/associate them with theorems. This applies to almost all of the math I know. Maybe this is a flaw in my own math abilities, but it seems like a lot of my colleagues also don't quite have a full understanding of the math they are using, even though they are really good at using it. Is this a common occurence? I want to get into theoretical physics, and I'm wondering if I should load up on more theoretical math courses, or if a really good working knowledge is sufficient for getting into the deeper theoretical stuff. I really do understand what I'm doing, I just don't know how to show it/explain it to others which strikes me as a problem in the future.

I know i'm posting this on the math forums and am likely to get a somewhat biased answer, but how much of a role does deep theoretical knowledge of math play in physics?