Green's Theorem is a neat theorem in that it relates a double integral over a region in the plane to an integral of a vector field on its boundary. One of my favorite applications is using it to find area, and in particular the area of an ellipse can be calculated very easily this way. On the other hand, the ellipse area has a good conceptual explanation in terms of the change of variables formula, and it doesn't really need Green's Theorem.
This leads me to my main question. Are there any interesting applications of Green's Theorem which no simpler proof is known?
An interesting application is to derive the Euler's equation, in the case of several variables, i.d. find out a necessary condition for a weak extrema of the following functional:
$$J[z]=\iint_D F(x,y,z,z_x,z_y) \; \mathrm d x \mathrm d y,$$
where $z=z(x,y)$ is the incognite surface, $z_x$, $z_y$ its partial derivatives, and F is continuously twice differentiable w.r.t. all its arguments.
The first variation of such a functional is,
$$\delta J = \iint_D \big (F_z h+F_{z_x}h_x+F_{z_y}h_y \big ) \; \mathrm d x \mathrm d y,$$
where $h(x,y)$ is an arbitrary increment which vanishes at the boundary $\partial D$.
Next, we observe that
$$\iint_D \big (F_{z_x}h_x+F_{z_y}h_y \big ) \; \mathrm d x \mathrm d y=\iint_D \frac{\partial }{\partial x}(F_{z_x}h)+\frac{\partial }{\partial y}(F_{z_y}h)\; \mathrm d x \mathrm d y-\iint_D \bigg ( \frac{\partial }{\partial x}F_{z_x}+\frac{\partial }{\partial y}F_{z_y} \bigg ) h \; \mathrm d x \mathrm d y.$$
Now, applying the Green's theorem to the first term on the right hand side
$$\int_{\partial D} F_{z_x}h \; \mathrm d y - F_{z_y}h \;\mathrm d x -\iint_D \bigg ( \frac{\partial }{\partial x}F_{z_x}+\frac{\partial }{\partial y}F_{z_y} \bigg ) h \; \mathrm d x \mathrm d y$$
The integral along $\partial D$ is zero, since $h(x,y)$ vanishes on the boundary of D. Hence comparing this last result with the increment, we find that
$$\delta J = \iint_D \bigg (F_z -\frac{\partial }{\partial x}F_{z_x}-\frac{\partial }{\partial y}F_{z_y} \bigg ) h(x,y) \; \mathrm d x \mathrm d y.$$
Thus, the condition $\delta J = 0$, and the fundamental lemma calculus of variations, imply the Euler's equation.