What's the connection between derivatives and boundaries?

The (second) fundamental theorem of calculus says that

$$\int_a^b f'(x) dx = f(b) - f(a)$$

which can also be stated, if one knows enough about what's coming next, as:

The integral of the derivative of a function over an interval is the same as the function evaluated at the (signed) boundary of the interval.

where I had to insert the word 'signed' to make it clear that there's an implicit multiplication by $-1$ when you evaluate the function at the 'bottom' end of the integral. If we wrote the right-hand side of the expression as

$$f(b) + (-1) f(a)$$

then even a high-school student could probably be persuaded that this is the same as 'integrating' $f$ over the two points $b$ and $a$, with a multiplication by $-1$ attached to the evaluation at $a$.

The generalization of this is the generalized Stokes theorem:

$$\int_C dw = \int_{\partial C} w$$

where $w$ is a differential form, $d$ is the exterior derivative, $C$ is a manifold on which $dw$ is defined, and $\partial$ is the boundary operator, which maps a manifold $C$ to its boundary.

This can be made to look pretty suggestive by writing integration of a form over a manifold using inner product notation:

$$\langle C, w \rangle \equiv \int_Cw$$

in which case Stokes' theorem becomes

$$\langle C, dw \rangle = \langle \partial C, w \rangle$$

which looks suspiciously like $\partial$ is the Hermitian adjoint of $d$.

But is that really the case? Differential forms and manifolds seem pretty different to me. If they are, in fact, related in this way, is there a theory which expounds upon this relation, generalizes it, or puts it in context with other areas of mathematics?

Solution 1:

Maybe this is the theory that you mean:

A manifold $M$ of dimension $m$ defines a $m$-current $[[M]]$, which is a functional on the space of smooth $m$-form in the following sense: $$[[M]](\omega)=\int_M\omega.$$ If $M$ is a manifold with boundary $\partial M$, then by Stoke's theorem the $m$-current $[[M]]$ and the $(m-1)$-current $[[\partial M]]$ is related by $$[[M]](d\omega)=\int_Md\omega=\int_{\partial M}\omega=[[\partial M]](\omega)$$ for any smooth $(m-1)$-form.

Personally, I first learned the theory of current from the lecture note of Demailly, which is available here.

Solution 2:

You are looking at homological currents. Like Thomas mentioned in a comment, you should consult some textbooks on geometric measure theory. (The recent introductory book by Lin and Yang seems fairly accessible.)

A bit more discussion of related ideas can be found at this MathOverflow thread.