Where is Greens theorem used? Where is Greens theorem used?
I think it's weird going from a vector field to calculating a volume on a scalar field, where do we use this kind of calculation?
 A: Perhaps one of the simplest to build real-world application of a mathematical theorem such as Green's Theorem is the planimeter. It's actually useful and extremely cool. 
Of course, Green's theorem is used elsewhere in mathematics and physics. It is a generalization of the fundamental theorem of calculus and a special case of the (generalized) Stokes' Theorem. Stokes' Theorem is the most general fundamental theorem of calculus in the context of integration in $R^n$. The fundamental theorem of calculus in $R$ says (under suitable conditions) that $\int_a^bf(x)dx=F(b)-F(a)$. Green's theorem is the analogue of this theorem to $R^2$.
A: One (complex-world) application of Green's theorem is in the proof of Cauchy's theorem, which states that for complex-valued function $f$ that is analytic inside and on a simple closed curve $\gamma$, $\int_{\gamma} f(z)\,dz = 0$. This is an extremely useful result from complex analysis. This shows us that Green's theorem is perhaps farther reaching than you might think, because we're using a result originally stated in terms of vector fields and scalar fields to show something about the complex numbers.
Green's theorem is actually a special case of a more general phenomenon, relating an integral over some surface to an integral over its boundary. The general theorem I'm alluding to is called Stokes' theorem, and it is extremely useful: sometimes we come across an integral over some surface that is very hard to evaluate, but using Stokes' theorem, we can change that integral into another one (which sometimes becomes much easier to evaluate!). This is actually a generalization of the Fundamental Theorem of Calculus: if $F$ is an antiderivative of $f$ on $[a,b]$, then $\int_a^b f = F(b) - F(a)$. The left hand side is the integral over the surface (in this case, just an interval), and the right hand side is the integral on its boundary (with the proper interpretations, of course).
A: As Stahl pointed out, Green's theorem is Stokes theorem in disguise for a 2D region.
Notice we can rewrite Green's theorem in 2D
$$
\oint_{\partial U} (Q\, dx + P\, dy) = \iint_{U} \left(\frac{\partial P}{\partial x} - \frac{\partial Q}{\partial y}\right)\, dx\, dy
$$
as 
$$ 
\oint_{\partial U} F\cdot n\,ds= \iint_{U} \nabla \cdot F\, dx\, dy
$$
for $F = (P,-Q)$. 
This form is powerful in that, we can exploit its "integral by parts" nature.
Let $F = \psi\nabla \phi - \phi\nabla \psi$, we can get Green's second identity (in two dimension):
$$
\iint_U \left( \psi \Delta \phi - \phi \Delta \psi\right)\, dx\,dy = \oint_{\partial U} \left( \psi {\partial \phi \over \partial n} - \phi {\partial \psi \over \partial n}\right)\, ds. \tag{1}
$$
Identities derived from Green's theorem like above play a key role in reciprocity in electromagnetism, the entry in wikipedia has a lot of examples. 
Some real life applications include using the reciprocity to evaluate the excitation from an impulse in waveguide or antenna designs.
In short, identity (1), which is derived from a generalized version of Green's theorem, allows certain types of reciprocity, which is a nature law, to be explained by a mathematical theory. 
