Editing Green's theorem (section)

==Relationship to Stokes' theorem==
Green's theorem is a special case of the [[Stokes' theorem#Theorem|Kelvin–Stokes theorem]], when applied to a region in the <math>xy</math>-plane.

We can augment the two-dimensional field into a three-dimensional field with a ''z'' component that is always 0. Write '''F''' for the [[Euclidean vector|vector]]-valued function <math>\mathbf{F}=(L,M,0)</math>. Start with the left side of Green's theorem:
<math display="block">\oint_C (L\, dx + M\, dy) = \oint_C (L, M, 0) \cdot (dx, dy, dz) = \oint_C \mathbf{F} \cdot d\mathbf{r}. </math>

The Kelvin–Stokes theorem:
<math display="block">\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \nabla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS. </math>

The surface <math>S</math> is just the region in the plane <math>D</math>, with the unit normal <math>\mathbf{\hat n}</math> defined (by convention) to have a positive z component in order to match the "positive orientation" definitions for both theorems.

The expression inside the integral becomes
<math display="block">\nabla \times \mathbf{F} \cdot \mathbf{\hat n} = \left[ \left(\frac{\partial 0}{\partial y} - \frac{\partial M}{\partial z}\right) \mathbf{i} + \left(\frac{\partial L}{\partial z} - \frac{\partial 0}{\partial x}\right) \mathbf{j} + \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \mathbf{k} \right] \cdot \mathbf{k} = \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right). </math>

Thus we get the right side of Green's theorem
<math display="block">\iint_S \nabla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \, dA. </math>

Green's theorem is also a straightforward result of the general Stokes' theorem using [[differential form]]s and [[exterior derivative]]s:
<math display="block">\oint_C L \,dx + M \,dy
= \oint_{\partial D} \! \omega
= \int_D d\omega
= \int_D \frac{\partial L}{\partial y} \,dy \wedge \,dx + \frac{\partial M}{\partial x} \,dx \wedge \,dy
= \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \,dx \,dy.</math>