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Recently, I've been doing some relatively more advanced math (multivariable and vector calculus) and the derivations of formulas are becoming increasingly complicated. I don't like using a formula if I don't understand why it works and where it comes from, so I have understood the derivation of all formulas I've used so far. My question is: is it hopeless for me to keep learning derivations? Will I eventually reach a point where understanding why the formulas I use work becomes too time-consuming and unreasonable? To clarify, I prefer to understand proofs in order use them, I don't memorize them. Any help is appreciated. Thank you.

John Doe
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  • Are there any other tags I can add to this? None other than advice came to my mind. – John Doe Nov 01 '20 at 10:25
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    Never a hopeless thing ! It is worth doing. – Spectre Nov 01 '20 at 10:25
  • Only if you understand and study will you remember things better; practice will help your hone your understanding of the topic. – Spectre Nov 01 '20 at 10:25
  • @Spectre It has sure seemed that way for a while... But the derivations are becoming so complicated and I haven't even started differential equations yet. So I have no idea how hard the can become – John Doe Nov 01 '20 at 10:26
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    I think you are right when saying you can't do maths without understanding what you're using. The converse is a lost of time: never try to do maths if you don't know what you are doing and don't understand what you are using. Keep practicing! It is worth it – Didier Nov 01 '20 at 10:27
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    @DIldier and Spectre Thank you for the advice. I guess I'll just keep trying, no matter how hard it gets. – John Doe Nov 01 '20 at 10:28
  • @JohnDoe , may I put up my advice as an answer ? – Spectre Nov 01 '20 at 10:34
  • @Spectre Of course! – John Doe Nov 01 '20 at 10:35
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    Should you ever feel the need to use that $a^n+b^n=c^n$ has no non-trivial integer solutions with $n>2$, or that every planar map can be $4$-coloured, then you may rest assured that your steps are fully justified, but you may prefer to not try to fully understand these used results. – Hagen von Eitzen Nov 01 '20 at 10:38
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    “Whether you think you can, or you think you can't--you're right.” - Henry Ford – DavidW Nov 01 '20 at 10:44

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First of all, you need very much of focus , without which all that you grasp may be lost eventually. Focus your mind when attending lectures, lessons or seminars. Keep a sharp ear.

Next, practice - deriving something once needn't always help you fix stuff in mind, so practising it as many times as you can fixes whatever you are learning in your mind.

Third - a strong heart. Never feel disheartened by the derivations you have to learn, be it complex or way above your level. Just give it a go, try again if you don't get it. Also, you can discuss it in here (I mean, the difficulties you find in trying the derivations and such stuff). As the lines in Bastille's 'Laura Palmer' goes - "Walking out into the dark, cutting out a different path, led by a beating heart..." - keep it beating, keep it brave, keep it right and you are ready to face the walls ahead of you. Never lose hope or faith in yourself.

Fourth - basics. I know you have the basics, but still it is inevitable that I say that anyone must have at least a small bit of the basics to get started.

Spectre
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Personally I don't think this is reasonable at all. You can easily spend 1 hour - 2 hours (or more in some cases) to fully understand a proof, and in multivariable calculus and linear algebra textbooks there might be a proof every 3 pages depending on how advanced the text is. In a Real Analysis text there might be a proof 2 pages. My point is, you should learn the proof if the theorem is important.

Sigma
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