Hey guys, I'm in some serious trouble - I don't know how to solve a lot of problems in electrostatics. The main reason is that once I seen that you had to be able to find electric fields & electric potentials due to charge distributions on quadric surfaces I panicked & gave up - I was still having trouble with finding the center of mass & moment of inertia of every quadric surface, every geometry from every location of the surface (just to make it hard on myself!) but I just felt there had to be a better way of dealing with these things than geometric considerations, ever prone to mistakes, but didn't know what to do, so I went off & learned a ton of maths... I had a similar problem with multiple integrals, once I seen the derivation of the area element in polar coordinates or volume elements etc... I just instinctively rejected the idea of polar integration, & would have seriously suffered had I not come to terms with mindless jacobian transformations & the curvilinear differential as a means to derive those things on the spot. During the year I finally learned how to use the inertia tensor & found that I could deal with every moment of inertia problem trivially compared to what I thought I'd have to do, exactly analogous to the way you can find area & volume elements for any change of variables with a jacobian & no case-specific geometric contortions, just plugging in formulas & being a bit careful about limits of integration. I'm wondering whether Laplace & Poisson equations can be used to derive all those formulas in an elementary calculus-based physics book? I'm talking Halliday-Resnick level, I would much prefer to be able to derive every formula myself without worrying about geometry too much (beyond setting up boundary conditions in a pde) & then using all those elementary geometric derivations as little exercises as opposed to my main source! What am I to do? I have a feeling that the derivations of moments of inertia in elementary physics books are done to apparently make it simpler on students who can't deal with tensors & multiple integrals, but for me it only made things immensely harder. I'm hoping Laplace & Poisson's equations will function as the electrostatics version of the inertia tensor, in other words I'm hoping a bit of advanced mathematics will make my life simpler - am I right, is there a source that would go through the common geometries using Laplace & Poisson equations to derive what's in those elementary physics books, or do I really have to go though (by my count) 37 different cases (including intersections of surfaces etc...) using error-prone geometric arguments just to feel I can deal with the electrostatic charge distributions & the electrostatic potential? Also, could you do all this with ease using differential forms? Thanks!