Re:Differential Forms and Integration (2008) [pdf]
by JadeNB on Friday May 22, 2020 @11:50AM
(#23273361)
> This list includes all particular cases of vector fields and differential forms of all orders on R^3 and on its sub-manifolds.
I think that first 'all' shouldn't be there, right? That is, these are all possible orders of vector fields and differential forms (among which the various Hodge dualities permit lots of identifications) on submanifolds of ℝ^n, but they're not all the possible particular cases of such vector fields and differential forms, in the sense that there are plenty of other, different physical situations that lead to the same mathematics (which, as you argue, is why the concept is so useful).
Re:Differential Forms and Integration (2008) [pdf]
by enriquto on Friday May 22, 2020 @11:58AM
(#23273456)
I meant all in terms of mathematical models. I think I did not forget any case. In R^3 you have p-forms for p=0,1,2,3 and vector fields; those are respectively cases (1), (5), (8), (2), (4) above. Of course each "case" may have several different physical interpretations that are modeled by the same mathematical object.
This concern means that you are looking at it backwards. Often, it happens in the opposite sense. You will find some problems that are hard to model or understand. Then, you realize that with quite a lot of effort you might be able to tackle them. And then, you learn about differential forms and see how they allow to express your problem very clearly and its solution becomes sort of immediate.
There is nothing mysterious about differential forms from the point of view of physics, but pure math texts often take this intuition for granted. If you are used to working with scalar and vector fields in space , you may realize that there are different kinds of each:
Examples of scalar fields:
(1) a potential (2) a density
Examples of vector fields:
(3) a velocity field (4) a flow (5) the gradient of a substance (6) the field of normal vectors on a surface (7) the field of tangent vectors to a curve (8) a field of "surface elements" filling the whole space
This list includes all particular cases of vector fields and differential forms of all orders on R^3 and on its sub-manifolds.
If you know the least amount of physics, you will realize that you can do some kind of integrals on these objects, but not all of them. For example, in the case of scalar fields, you can integrate a density over a domain, or you can evaluate a potential at one point (or more often, the difference of potential between two points). Thus, potentials are 0-forms and densities are 3-forms. And a similar reasoning for the vector fields, and 1-forms and 2-forms.