Physics Calculations That Apply to a Variety of Geometric Objects

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I had a serious problem when I read my first book on calculus-based physics when I came to the center of mass & moment of inertia sections of my books. I really freaked myself out over the fact that even restricting ourselves to the most common & ideal geometric objects the list of things you'd need to be able to compute the COM & MOI for would be huge & it seemed to me that every physics book took a few slightly different scattered examples. What made matters worse was that all the different books I looked in seemed to give different derivations of the same concepts & compounding the complexity was the fact that calculus books seemed to do things in just a completely different manner to the physics books.

It took me a quite a while to realize & accept that the derivations & variations of derivations in all the books basically reduce to applying a mish-mash of single, double, triple & surface integral techniques arbitrarily combined with the geometry of the situation arising & leaving in the derivations at random stages & that the kinds of geometric objects you can apply these techniques to are n-gons, conics, quadric surfaces let alone random portions, modifications (hollow, full etc...) & combinations of all of these (which is no small list when expanded out)...

Based on this little motivation, my question is: What other things in physics are there that once you encounter them you need to be able to calculate them for absolutely every geometric object I've mentioned? If it's not clear what I'm looking for, good answers could be: radius of gyration, gravitational field due to geometric objects, charge density of geometric objects containing charge, electric field due to geometric objects containing charge via Coulomb's Law, via Gauss law, electric potential due to geometric objects, hopefully you see what I mean...

Lest you think this an idle question, it's really functioning for me as a means to ensure I can deal with, & prepare for, certain aspects of advanced electromagnetism, quantum mechanics, statistical physics, classical & quantum field theory & be aware of the things that would hold me back in learning them. Had I been aware of this way of looking at things before I ever tried to learn electromagnetism I'd have breezed through the books, skipping these things has resulted in a sincerely painful headache of false knowledge & I never want to go through that again. I can't tell you how much I'd appreciate a helpful answer to this question so if you have a moment I'd be really greatful, thanks for your time

 

[AlephZero]

What other things in physics are there that once you encounter them you need to be able to calculate them for absolutely every geometric object I've mentioned?

To a first approximation, "all of them, for any conceivable geometry, not just the special cases you listed".

The problems in physics textbooks and homework questions are chosen to be simple enough so you can solve them by hand. The real world isn't like that. Back in the 1940's, the wings of the WWII Spitfire figher aircraft were designed as a semi-ellipse, because the designer couldn't calculate the aerodynamics of anythng more complicated (and luckily a semi-ellipse happens to be a pretty good shape to choose). But that was 70 years ago.

Nobody does these calculations for real-world geometries by evaluating integrals by hand any more. That's what computers and numerical methods are for!

 

[press]

Do you mean you'd rather the whole bouquet of redundant theorems/definitions/methods be introduced under an unified umbrella theorem/definition/method? That is, to have only one definition to apply to solve/derive million other definitions/problems and have only handful of those to cover pretty much all of physics. For that you probably need mathematical x-ray vision that sees the skeleton of the problem immediately and sorts the chaff out automatically, only leaving the central point. Sounds cool.

 

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To a first approximation, "all of them, for any conceivable geometry, not just the special cases you listed".

Well before reading on, just think about something like quantum mechanics & how you think a subject like that would have the kind of calculational aspects to it that my thread is asking about. I have one brilliant example that I should have mentioned in my original post so hopefully this will be a good indication of what I'm looking for, though I hope you'll think about it yourself before reading this:

I'd mention the fact that the Laplace, Poisson, Helmholtz & Schrodinger equations are separable in 11 coordinate systems (if they satisfy certain conditions) & that you could pierce through the structure of a wealth of examples involving PDE's in physics by recognizing the separability criterion involved. Note this simple fact gives you the power to take a textbook example & modify it in tons of different ways to apply to different geometries, & you can see instantly why some approaches are doomed to fail & can justify it as opposed to just getting stuck & having a headache. It's actually crazy that I'd think of this example because if you think about it it's "all derived from confocal quadrics" (to quote the introduction of that book). In other words this example is indirectly & implicitly linked to the examples in my OP.

I'm hoping people with experience would contribute any little pieces of intuition they've picked up no matter how small

(Also, interesting ellipse example)

Do you mean you'd rather the whole bouquet of redundant theorems/definitions/methods be introduced under an unified umbrella theorem/definition/method? That is, to have only one definition to apply to solve/derive million other definitions/problems and have only handful of those to cover pretty much all of physics. For that you probably need mathematical x-ray vision that sees the skeleton of the problem immediately and sorts the chaff out automatically, only leaving the central point. Sounds cool.

Well I think the separability example I gave above unifies under a ton of examples, if we accumulate a lot of these examples in different fields there might be some structure underlying all these substructures in the way that quadric surfaces seem to underlie all the examples I've given thus far