Edit: I just noticed that the exercise itself was asked here. However I'm not asking for a proof, but about the meaning/intuition behind the notation / exercise itself.
On a Multivariable Calculus problem sheet I was asked to prove the following integral identity:
Problem. Let $f$ be a smooth scalar field defined on a region $R \subseteq \mathbb{R^3}$ with a smooth boundary $\partial R$. Show that $$\iint_{\partial R} f \mathbf{r} \wedge \text{d} \mathbf{S} = \iiint_R \mathbf{r} \wedge \nabla f \, \text{d}V$$
I did manage to solve the problem by using the fact that d$\mathbf{S}$ = $\mathbf{n}$dS, where $\mathbf{n}$ is the outward-pointing normal to $\partial R$ at $\mathbf{r}$ and by applying the divergence theorem on a well-chosen vector field (I won't go into details in case this problem appears on a problem sheet in the future.)
I have two questions about this problem:
1) Is there some sort of meaning behind taking a cross product with the area differential, or is it in this case just a shorthand notation for $\mathbf{r} \wedge \mathbf{n}$ dS? I was very confused when I first saw it, and it seems like it's not really necessary, but why would they choose this notation if it doesn't really mean anything special?
2) Is there any kind of (perhaps physical or otherwise) interpretation of this integral identity, or is it just the result of some arbitrary calculation? I can't wrap my head around what it's supposed to mean...