When I tried to think of how I'd answer this question, I realized that never in my undergraduate curriculum was I asked to compute the surface or line integral of a vector field. I don't mean I've never been asked to compute the flux or circulation of a field (meaning the field dotted with the surface normal, in the case of a surface integral, or the field dotted with a tangent vector, in the case of a line integral). I mean I've never been asked to compute things like $$\int_Sfd\mathbf{S}, \int_S \mathbf{f}dS, \int_c\mathbf{f}\times d\mathbf{r}$$ (where vectors are bolded and scalars aren't). I have two questions:
- In what contexts do such integrals -- integrals of vector fields that yield a vector rather than a scalar -- arise?
- Why are integrals like these pretty much never encountered in a standard course in undergraduate vector calculus? (I cannot think of a textbook where I could find them. If you can, please mention it.)
NOTE: I'm not asking how to do these integrals; I realize you can just compute them component-wise.
These multidimensional scenarios arise in pdes, take for instance the navier stokes equations:
$\rho(\frac{\partial v}{\partial t}+v\cdot\nabla v)-\mu\Delta v+\nabla p=f(x)$
This system of equations in defined in $\Bbb R^n$, and it's derivation relies on the integration of vector fields.