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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:

  1. In what contexts do such integrals -- integrals of vector fields that yield a vector rather than a scalar -- arise?
  2. 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.

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  • $\begingroup$ just write vectors in it's components and you would get normal surface integrals for each components. $\endgroup$
    – S L
    Commented May 12, 2014 at 18:31
  • $\begingroup$ I didn't ask how to do them. I understand how I would do them. $\endgroup$ Commented May 12, 2014 at 18:33
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    $\begingroup$ I was posing myself more or less the same question. While vector-valued integrals arise sometimes they are a lot less frequent. Must have something to do with differential geometry. Indeed, while circuitation integrals and flux integrals surely have some invariant interpretation, vector valued integrals probably do not. (A place where one might look for the answer is Frankel's book The geometry of physics). $\endgroup$ Commented May 12, 2014 at 18:36

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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.

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