I recently solved the following task:
Let $A = [0,1]^3$ and $\omega = \dfrac{x_1^2 x_2^3}{1+x_3^2} \ dx_1 \wedge dx_3$ Show that this fulfills stokes theorem by showing that $\displaystyle \int_A \ d\omega \ = \ \displaystyle \int_{\partial A} \ \omega $.
That worked out really well. As the solution I got for both sides $-\dfrac{\pi}{12}$. The question is how to interpret this (maybe physically). I just learned about diffrential forms a few days ago and Im not sure how to interpret them. I think that integrating a 1-form over a curve gives the work that the vectorfield applies on a particle that walks along this curve. Is $-\dfrac{\pi}{12}$ something like the work that the vectorfield applies on the cube or something like that? Please keep in mind that Im not a physicist at all.
OK, so the physical interpretation is that you're finding the flux of the vector field $\vec F = \begin{bmatrix} 0 \\ -x_1^2x_2^3/(1+x_3^2) \\ 0\end{bmatrix}$ outwards across $\partial A$. In this setting, $d\omega = (\text{div}\, \vec F) dx_1\wedge dx_2\wedge dx_3$, and $\text{div}\,\vec F$ tells you infinitesimally whether you have a source or a sink (or neither) at each point; adding these up all over $A$ gives the net flux across the boundary.