Doubts on Green's Theorem

Question

I'm trying ro understand Green's Theorem, but got stuck on some- maybe simple- problems. What exactly are C and D (see the formula below)? Knowing this, when should I use Green's Theorems? In what kind of problems it would be useful? How? All answers are welcome. $$\oint_C Pdx+Qdy=\iint_D\left(\frac{\partial Q} {\partial x}-\frac{\partial P} {\partial y}\right)\,dA $$

Answer 1

$\textbf{Differential Forms Approach}$: Green's theorem is just a consequence of Stokes' theorem.

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

where $\omega \in \Omega^{n-1}(M)$. In our case, $D$ is our "nice set" (not necessarily a manifold), $\partial D = C$ and $\omega = P dx + Q dy$ and so,

$$d\omega = d(P) \wedge dx + d(Q) \wedge dy = \frac{\partial P}{\partial y} dy \wedge dx + \frac{\partial Q}{\partial x} dx \wedge dy = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \ dx \wedge dy$$

Since $(x,y)$ are the coordinates on $ \mathbb{R}^2$ then $dA = dx \wedge dy$. In most contexts, $D$ will be an open set or a closed set in which the integrand vanishes and $\partial D$ means the boundary of $D$. To answer your question about when can you use it, well your vector field has to be $2$-dimensional or in differential forms language, $\omega$ needs to be a $1$-form.

Let me know if anything here needs to be cleared up. It is rather difficult to give a precise definition of manifold-boundary without some high-powered language, but points in the boundary for a plane region are those in which neighborhoods look like the upper half of an open disk.

$\textbf{Edit}$: Other users have made a really good point. There is no reason why I should have to use differential formats explain this. My intention was to give you a statement in terms of differential forms that helps me remember the conditions for Green's Theorem. The fact that we just need $\omega \in \Omega^1(M)$ means $\omega = F \cdot d \textbf{r}$ and since the exterior derivative of $1$-forms is just the curl,we have $\textrm{curl}(F) \cdot d\textbf{S} = d\omega = d(F \cdot \textbf{r})$.

$$\int_{\partial D} \omega = \oint_C F \cdot d \textbf{r} = \int_D d\omega = \int_D \textrm{curl}(F) \cdot d\textbf{S}$$