In classical mechanics you want to calculate the moment of inertia for hollow & solid:
lines, triangles, squares/rectangles, polygons, planes, pyramids, cubes/parallelepiped's, circles, ellipses, parabola's, hyperbola's, sphere's, ellipsoid's, paraboloid's, hyperboloid's, cones & cylinder's
setting them up in either scalar notation (i.e. geometrically deriving it as in elementary physics books) or in tensor notation, which at least for me is an immense task with so many possibilities I still haven't fully carried it all out.
My question is, how does all this translate over the special &/or (?) general relativity? Do you have to re-do every one of those calculations from a more general standpoint or is it just that the density in the integral is usually variable?
As a side note, is there an easier & more way to do all of the above?
For instance, in calculus books they sometimes put MoI into 3 different chapters, leading to single, double & triple integral modelling on top of physical modelling (in physics books) or tensor modelling (in advanced physics books) which is really 5 ways to set up about 30 geometric models...
Thinking about it, you can apparently sometimes use Stokes theorem, e.g. slide 26, to show some of these models are exactly equivalent, but how do I deal with it all in general in a unified manner, where do I start to derive it all in as simple a manner as possible? Thanks