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I'm interested in finding a elementary(ish) resource on vector-valued integrals. By vector-valued integrals I mean integrals of the form $\int_D \mathbf f\,dS$ which would yield a vector quantity as opposed to the usual $\int_D \mathbf f \cdot d\mathbf r$ or $\int_D f(\mathbf r) ds$, etc that yield scalar quantities.

I don't remember covering any vector-valued integrals in multivariable calculus, though I used the specific integrals $\int \mathbf a(t)dt = \mathbf v(t) - \mathbf v(0)$ and $\int \mathbf v(t)dt = \mathbf r(t) - \mathbf r(0)$ somewhat in physics.

In particular, I'd like to learn any relevant integral theorems for these types of integrals, as all the usual integral theorems (Stokes', divergence, Green) are theorems about scalar-valued integrals. I googled vector-valued integrals but I was getting very complicated looking stuff with measure theory which I don't yet understand. Is there a simpler reference for these types of integrals?

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    (1) In some sense, vector-valued integrals aren't all that different from scalar integrals; if $\mathbf{f} = f_x \hat{x} + f_y \hat{y} + f_z \hat{z}$, then $\int_D \mathbf f\,dS = \left[ \int_D f_x \,dS \right] \hat{x} + \left[ \int_D f_ \,dS \right] \hat{y} + \left[ \int_D f_z \,dS \right] \hat{z}$. The usual scalar integral theorems apply to each of the quantities in this equation. – 2017-02-20
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    (2) If you do want to write this in a more compact notation (or write down versions that work in non-Cartesian coordinate systems), then the natural generalizations of things like the divergence theorem involve *tensor calculus*. I'm not sure what the best pure math reference for this subject is, but knowing the terms might give you something to search on. (I'm mainly familiar with the subject from its applications to physics, particularly general relativity.) – 2017-02-20
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    Cool. Thanks for the info! – 2017-02-20
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    I asked almost the same question [here](http://math.stackexchange.com/questions/792026/integrals-of-vector-fields-that-yield-vectors-not-scalars). Many books (e.g. Apostol's volumes on multivariable calculus) mention $\int_D \mathbf{f}\,dS$ for $D$ an interval in $\mathbb{R}$ (where $dS=dt$), pointing out that the integral is carried out component-wise but not generalizing to higher-dimensional domains. Indeed I haven't found a standard multivariable calculus reference that discusses $\int_D \mathbf{f}\,dS$ when $D$ is a domain in $\mathbb{R}^2$. – 2017-03-05
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    @symplectomorphic Apparently they show up in differential geometry, so when I get there (eventually) I'll let you know. – 2017-03-05
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    Well, integrals like $\int_D \mathbf{f}\,dS$ where $D\subset\mathbb{R}^2$ or $\mathbb{R}^3$ show up all the time in electromagnetism; I first encountered them in the first two chapters of Griffith's *Introduction to Electrodynamics*. They are also discussed in Arfken's *Mathematical Methods for Physicists* under the heading "Vector Integration." Still, it is curious that standard courses for math majors don't draw these connections. – 2017-03-05

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Unless I misunderstand your question, you $should$ have covered vector-valued integrals in Calculus II. It is very basic as you simply integrate each component function separately. In this sense it is like differentiating a vector-valued function (which is also componentwise). On the other hand, if you were to increase the dimension of the domain beyond 1, then integration and differentiation become interesting enough that the inherent linear algebra becomes more apparent.

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    How to do it in Cartesian coordinates (componentwise) might have briefly been covered, but I was looking for a more general coordinate system-agnostic theory. Also, yes, I'm interested in domains of dimension greater than $1$. – 2017-02-20