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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?

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Suggestion: have a look at text books on Geometric Measure Theory. – user20266 Jan 5 '12 at 13:06
Speaking as a physics major, your suggestive way of expressing Stokes' theorem is just about the most mesmerizing formula I've ever seen. – David H May 29 '13 at 22:54
up vote 14 down vote accepted

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.

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Ah, beat me by a sec! – Willie Wong Jan 5 '12 at 14:04
@Willie: haha. We mean essentially the same thing. – Paul Jan 5 '12 at 14:06

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.

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Let's assume your manifold $C$ with boundary $\partial C$ is given as a subset of a manifold $M$ without boundary. Let $\chi_C:M\mapsto\mathbb R$ be the characteristic function of $C$, i.e. $\chi_C(x)=1$ for $x\in C$ and $\chi_C(x)=0$ for $x\notin C$. Then clearly $$\langle C, w \rangle \equiv \int_Cw=\int_M\chi_Cw$$ Now let's look at $d(\chi_Cw)$. If $\chi_C$ would be a smooth function, we would have $d(\chi_Cw)=d\chi_C\wedge w+\chi_C\;dw$. Now let's interpret $d\chi_C$ in the sense of distributions such that $\int_Md\chi_C\wedge \phi=-\int_{\partial C}\phi$, then we can conclude $$\langle C, dw \rangle = \langle \partial C, w \rangle$$ from $\int_Md\phi=0$ for $\phi$ with compact support.

A totally different generalization of the fundamental theorem of calculus would be related to the joint cumulative distribution function $F(x_1,\ldots,x_n):=P(X_1\leq x_1,\ldots,X_n\leq x_N)$. This function $F$ is some sort of multivariable antiderivative, because it allows to compute $$\int_{a_1}^{b_1}\ldots\int_{a_n}^{b_n}f(x_1,\ldots,x_n)dx_1\ldots dx_n=\sum_{p_1\in\{a_1,b_1\}}\dots\sum_{p_n\in\{a_n,b_n\}}(-1)^?F(p_1,\ldots,p_n)$$ where $(-1)^?$ is either $1$ or $-1$, depending on whether the number of $p_i$ with $p_i=a_i$ is even or odd.

This formula can be generalized to allow integration over polygonal domains with only a finite (small) number of different angles.

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