Green's Theorem

A demonstration of how a large loop can be decomposed into a family of infinitesimal loops.

Quantifying "circulation density" is best introduced in 2 dimensions. Given a large counter-clockwise oriented loop $C$ that is confined to 2 dimensions, $C$ can be decomposed into a family of infinitesimal loops as shown on the right. Boundaries that are common to adjacent loops cancel each other out due to their opposite orientations, so the total circulation around $C$ is the sum of the circulations around each infinitesimal loop.

An infinitesimal rectangular loop.

Consider the infinitesimal rectangle $R=[x_{l},x_{u}]\times [y_{l},y_{u}]$ . Let $(x_{c},y_{c})\in R$ be an arbitrary point inside the rectangle, let $\Delta x=x_{u}-x_{l}$ and $\Delta y=y_{u}-y_{l}$ , and let $\partial R$ be the counterclockwise boundary of $R$ .

The circulation around $\partial R$ is approximately (the relative error vanishes as $\Delta x,\Delta y\rightarrow 0^{+}$ ):

$\int _{\mathbf {q} \in \partial R}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} \approx \mathbf {F} (x_{u},y_{c})\cdot (+\Delta y\mathbf {j} )+\mathbf {F} (x_{c},y_{u})\cdot (-\Delta x\mathbf {i} )+\mathbf {F} (x_{l},y_{c})\cdot (-\Delta y\mathbf {j} )+\mathbf {F} (x_{c},y_{l})\cdot (+\Delta x\mathbf {i} )$ $=F_{y}(x_{u},y_{c})\Delta y-F_{x}(x_{c},y_{u})\Delta x-F_{y}(x_{l},y_{c})\Delta y+F_{x}(x_{c},y_{l})\Delta x$ $=\left({\frac {F_{y}(x_{u},y_{c})-F_{y}(x_{l},y_{c})}{\Delta x}}-{\frac {F_{x}(x_{c},y_{u})-F_{x}(x_{c},y_{l})}{\Delta y}}\right)\Delta x\Delta y$ $\approx \left({\frac {\partial F_{y}}{\partial x}}{\bigg |}_{(x_{c},y_{c})}-{\frac {\partial F_{x}}{\partial y}}{\bigg |}_{(x_{c},y_{c})}\right)\Delta x\Delta y$ $\approx \iint _{R}\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)dxdy$

As $\Delta x,\Delta y\rightarrow 0^{+}$ , the relative errors present in the approximations vanish, and therefore, for an infinitesimal rectangle, $\int _{\mathbf {q} \in \partial R}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\iint _{\mathbf {q} \in R}\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)dxdy$

${\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}$ is the "circulation density" at $(x_{c},y_{c})$ . Let $C$ be a counter-clockwise oriented loop with interior $D$ . The circulation around loop $C$ is the total circulation contained by $D$ : $\int _{\mathbf {q} \in C}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\iint _{\mathbf {q} \in D}\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)dxdy$ . This is Green's theorem.