Differential form volume

Question

For a closed curve $C$ that bounds the region $B$ in the plane, we have that $\int_Cxdy-ydx=\int_B2dxdy=2vol(B)$. Are there similar formulas for 3 or more dimensions? Instead of $xdy-ydx$, what would the corresponding 2 form $\omega$ be such that $d\omega=dxdydz$?

Answer 1

What you have observed is Green's theorem in the plane can be used to write down a formula for the integral over any region bounded by a simple closed curve. The general case requires generalizing the concepts of vector calculus that lead to that formula, because essentially what you want to do is be able to say something like:

$$ \int_{\text{Region}} 1 dV = \int_{\text{boundary of that Region}} \text{Something } dS$$

The question is essentially how you should find that something. The fundamental theorem of calculus has the following generalization (the generalized Stoke's Theorem) for differential forms.

$$\int_M \ d \omega = \int_{\partial M}\omega$$

where $\omega$ is now a differential form and $M$ a manifold.

So the answer to your question then boils down to figuring out how to find a differential form $\omega$ so that $d\omega$ is the volume form. How to find these on manifolds can be difficult, if not impossible, as there is no guarantee the volume form actually is exact (in fact on compact manifolds without boundary it is never exact, by Stoke's theorem), but on $\mathbb{R}^n$ this is not so hard, as the formula for $d$ is very simple, and the volume form has a very easy to write down expression as $dx^1 \wedge ... \wedge dx^n$. In particular, if $\omega$ is a form, it can be represented on Euclidean space as $\sum u_I dx^I$, where $I$ is a multi-index. The expression for the exterior derivative is just

$$\sum_{k=1}^n\frac{\partial }{\partial x^k}\bigg(\sum u_I dx^I \bigg) \wedge dx^k$$

Then we can at least see from direct calculation that the form $$\omega = \sum_{k=1}^n (-1)^{k+1} x^k dx^1 \wedge ... \wedge \overline{dx^k} \wedge ... \wedge dx^n$$ will have the desired property, as taking $d$ term by term inserts the missing partial and is $0$ when terms are repeated.

Answer 2

You stated Green's theorem for the case $L dx = -y dx$ and $M dy = x dy$.

Its generalization is Stoke's theorem. A differential form for this is $\omega = x dy \wedge dz + y dz \wedge dx + z dx \wedge dy$