TITLE: Integration of forms and integration on a measure space
QUESTION [70 upvotes]: In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):

the indefinite integral $\int f$ (also known as the anti-derivative),
the unsigned definite integral $\int_{[a,b]} f(x) dx$ (which one would use to find area under a curve, or the mass of a one-dimensional object of varying density),
and the signed definite integral $\int _a^b f(x) dx$ (which one would use for instance to compute the work required to move a particle from a to b).

When one moves from single-variable calculus to several-variable calculus:

The indefinite integral generalises to the notion of a solution to a differential equation, or of an integral of a connection, vector field, or bundle.
The unsigned definite integral generalises to the Lebesgue integral, or more generally to integration on a measure space.
Finally, the signed definite integral generalises to the integration of forms.

While learning from this article, I tried to find the counterpart of the later two kinds of integration(as the title indicates) in the several-variable calculus I learned before. Now I am considering the following four kinds of integration:

Line integral of a scalar field
Line integral of a vector field
Surface integrals of scalar fields
Surface integrals of vector fields

Here are my questions:


What kinds of integration are these four ones according to the categories in the article? (I just guess generally the scalar one is the integration on a measure space and the vector one is the integration of forms.)

How do they belong to the category respectively? (For example, if it is the integration on a measure space, then what exactly is the underlying measure space?)

REPLY [12 votes]: The line and surface integrals of multivariable calculus can be extended in several different ways to more advanced and more general settings. So there isn't necessarily one correct answer to your question. Here is one way of putting these integrals in a more advanced setting, which will answer your "guess" above in the affirmative. I will assume throughout that we are concerned with integrals over subsets of $\mathbb{R}^n$, as in undergraduate calculus.
For integrals of scalar-valued functions on unoriented subsets of $\mathbb{R}^n$, one can use the Lebesgue integral with respect to $k$-dimensional Hausdorff measure $\mathcal{H}^k$. The line integral of a scalar function $f$ over a curve $C$ in $\mathbb{R}^3$ is then:
$$ \int_C f \, ds = \int_{\mathbb{R}^3} f \, d\mathcal{H}^1,$$
where I assume that $f$ is defined to be 0 off of $C$. This has the advantage of generalizing well in several directions. First, when integrating differential forms, you are usually restricted to differentiable manifolds of integer dimension. With the Hausdorff measure approach, you can integrate over sets that aren't necessarily differentiable submanifolds of $\mathbb{R}^n$ or over subsets of non-integer dimension (ex: fractals). Second, this definition makes (scalar) line, surface, and volume integrals in $\mathbb{R}^3$ all the same type of integral (just different dimensions of Hausdorff measure), since $n$-dimensional Hausdorff measure is equal to Lebesgue measure on $\mathbb{R}^n$.
(Disclaimer: While this definition is great in theory, it is fairly useless in practice. It doesn't give you any new way of calculating a standard line integral, for example.)
For integrals of vector-valued functions on oriented submanifolds of $\mathbb{R}^n$, one can use differential forms. I'll use $\mathbb{R}^3$ for the example. Consider the vector field $F=(f,g,h)$ on $\mathbb{R}^3$. We can associate two different differential forms with this field. We have the 1-form $\omega = f\,dx + g\,dy + h\,dz$ and the 2-form $\eta = f\, dy \wedge dz + g\, dz \wedge dx + h\, dx \wedge dy$. Then we find that
$$d\omega = (h_y-g_z)\, dy \wedge dz + (f_z-h_x)\, dz \wedge dx + (g_x-f_y)\, dx \wedge dy$$
which corresponds to $\operatorname{curl} F$, and we find that
$$d\eta = (f_x+g_y+h_z) \, dx \wedge dy \wedge dz$$
which corresponds to $\operatorname{div} F$. In this setting, the theorems of Green, Gauss, and Stokes from multivariable calculus are all just special cases of Stokes' Theorem:
$$\int_M d\omega = \int_{\partial M}\omega.$$
There is a little work involved in making sure that everything works out as it should. For example, in the Divergence Theorem (Gauss's Theorem) you need to check that if $M$ is a 3-submanifold of $\mathbb{R}^3$, then the 2-form that is induced from $\eta$ by the inclusion $\partial M \rightarrow \mathbb{R}^3$ corresponds to $F \cdot n \, dS$. But no worries, everything works out fine.
So to answer your question: yes, you can think of integrals of "scalar fields" as integration on a measure space and integration of "vector fields" as integration of differential forms (as long as you are considering the standard "Calc III" integrals over subsets of $\mathbb{R}^n$ with the standard structures inherited from $\mathbb{R}^n$). Just keep in mind that these aren't the only ways of thinking of these integrals, as you can see from Willie's answer.