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i am a soon to be graduate in physics and i am considering to do a master in the theoretical physics area, but i find myself a little bit "unfaithfull" of my mathematical skills.

Let me explain i don't have a perfect GPA, i'm on average on my faculty and what is pushing down my GPA are my grades in Analysis and Linear Algebra,which were my first exams when i started uni three years ago.

Even at the end of my bachelor i attended a differential geometry class and i found myself struggluing with exercises especially with the proof based ones.

Like i had a really hard time in trying to prove that a space wasn't Hausdroff or to show that two spaces were Homeomorphic or showing that a manifold is not orientable (these are just some examples) so what i am asking is: since i don't have difficulties in the computation part ( i mean i can do integrals,limits and excetera), what should i do to improve my "proofing" skills?

Should i study again my calclus books and algebra ones? I' ve a couple of months before starting the master so i could spend some time (a lot if it's useful) on reviewing the basics.

I feel like i am missing some sort of logical way to think when i approach this problems(also when i do these types of exrcises, the textbooks that i have never put solutions, so i don't know how to check if what i'm doing is correct).

Sorry for my poor language, english it is not my mother tongue, also sorry if this questions come a little weany, but i want to improve since at this point it's almost a professional matter

Thank you to anyone who will read and/or answer to this post.

Lip
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It is my opinion becoming proficient in writing proofs comes with practise. Speaking from my own experience, once one becomes comfortable with basic logic then all difficulty in writing proofs comes from trying to prove some statement in which one does not fully understand in the first place. Take for example the classical argument that the square root of 2 is an irrational number (see this post for details Proof that $\sqrt{2}$ is irrational). Once you are clear about the logic of a proof by contradiction, all one needs to understand this proof is the properties of rational numbers and some basic algebra. Far more complicated statements are proved using a proof by contradiction argument, so what's the problem ? The devil is in the details!

That being said, I would not recommend to spend time learning pure logic, this is something you will pick up as you go. I was never a fan of these "proof books" in which they are specifically written to help you learn "how to write proofs". To me, this is not an optimal way to spend your time, it is much better to read a rigorous book that is relevant to your field. You will become much more comfortable writing proofs and simultaneously have a deeper understanding on an area of mathematics that will benefit you for the rest of your career.

Thus what I would recommend is to pick an area of mathematics that you feel will be of great use to you as a theoretical physicist (I am not a physicist myself so I will refrain from giving an ill informed recommendation) and begin reading a rigorous introductory text. As you progress, proving all statements as you go, you will naturally start to see the common themes that appear and begin to realise that most proofs are essentially the same idea, the only difference being the statements you are trying to prove and the details needed.

Personally, I think the best area to choose is real analysis. Any half decent text on real analysis will get you well acquainted with proof writing.

I hope this helps.

Emmet
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  • Generally agree with this answer, except for one claim: “most proofs are essentially the same idea, the only difference being the statements you are trying to prove and the details needed.” Kind of like saying, most dishes are essentially the same, the only differences being the ingredients, the preparation, and the flavor. – Michael Weiss Mar 31 '23 at 20:42
  • Thanks for the comment. If I were to try and construct a bijection between my answer and your analogy, I think the dishes would be the final statements and the proofs would be the cooking (what we want is to get better at cooking). In which case I stand by my statement as in your analogy it amounts to staying for some dishes you may be chopping up carrots and boiling potatoes but with others you are chopping up peppers and boiling rice. The dishes and ingredients are very different, but in both cases all your doing is chopping and boiling! – Emmet Apr 01 '23 at 21:19
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    And with other dishes you'd be grating, or sauteeing, or wrapping in dough, or baking, or broiling, or whipping egg whites, or using heatless marinades (like ceviche), or... To my mind, your statement sells short the rich variety of mathematics. I will agree that there are key themes, and if you get good at doing proofs in, say, real analysis, much of that will carry over to other fields. (But a lot won't--e.g., residues in complex analysis, or the method of diagrams in model theory, or chasing diagrams in category theory, or...) Like I said, I agree with most of your answer. – Michael Weiss Apr 01 '23 at 23:20