I seldom see integration of vector fields over volumes or surfaces in pure math, but that practice seems to be quite widespread in physics. The integral of the momentum density in a volume $\Omega$ in $\mathbb{R}^3$, for instance, can be written as $$ \mathbf{P} = \int_{\Omega} \rho(\mathbf{r}) \mathbf{u} \mathrm{d} V $$, where $\mathbf{u}$ is a vector field (velocity field) and $\rho$ a scalar field (mass density). It is pretty clear to me how to perform this integration (componentwise basically), and the result is a vector( $\mathbf{P}$), but I can't seem to remember any maths textbook discussing or defining this type of integration. Is my memory just failing me or is this the case? Is there some problem with the integration of vector fields over volumes that I'm failing to grasp? Is there any reason why it would be uninteresting to mathmaticians?
I can see that perhaps the location of $\mathbf{P}$ is not so well defined (meaning, does it live on a tangent space $T$ and if so where exactly?) and I can't see 100% how it fits with differential forms and the like, perhaps there are issues with integrating this over more general things than $\mathbb{R}^n$? are these issues important enough that mathmaticians avoid this completely? In short: why is this avoided? (if it is).