tensor calculus - Differential Forms and Densities

I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a region of $3$-space, and let $\rho : U \to \mathbb{R}$ give the charge density at every point of $U$. I can then create the $3$-form $\omega = \rho \ dx \wedge dy \wedge dz$, which in my understanding gives me the approximate amount of charge enclosed by a volume determined by $3$ vectors when they're given.

So, if I give the vectors $v, u, w$, the value of $dx \wedge dy \wedge dz(v,u,w)$ should be the volume enclosed by those vectors, and hence $\omega(u, v, w)$ should be an approximation of the charge enclosed. Is this correct?

My only problem is: in this point of view, the form isn't giving me the density, the density itself is being given by a scalar field, while the form gives me the charge instead of the density.

Is this correct? The form is always meant to give the charge instead of the density? The density should always be regarded as a scalar field?

Answer

By 'density' in this case I think you just mean "something on a manifold that can be integrated to give you a scalar". By this definition, on an $n$-manifold, a density would be an $n$-form (since if you integrate over a form of lower dimension you get zero). So in your 3d case, take 3 smooth functions $f,g,h:M^3\to \mathbb{R}$, the form

$df\wedge dg\wedge dh$

is a density. Now, in your example you are integrating over a scalar field multiplied by a 3-form, which is again a 3-form, which can be integrated over a 3-manifold to get you the change in the region. But the scalar field $\rho:M^3 \to\mathbb{R}$ is NOT a density (not a 3-form), so it cannot be integrated over to find the total charge. The charge density is

$\omega=\rho dx\wedge dy\wedge dz,$

and $\rho$ just tells us how 'big' this should be.

In other words, the mathematical term 'density' can be stated as '$n$-form on $M^4$', whereas the colloquial 'density' for 'something per unit length/area/volume' is shorthand for what we really mean ($n$-form).