Green's Theorem in Vector Form
\[P,Q\]
are differentiable functions on a region \[R\]
in the plane with a boundary \[C\]
, then \[\oint_C P \:dx + Q \: dy = \int \int_R ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dx \: dy\]
Green's Theorem can be written in vector form. Let
\[P \mathbf{i} + Q \mathbf{j}\mathbf{a}\]
then the left hand side of Green's Theorem becomes
\[\oint_C \mathbf{a} \cdot d \mathbf{r}\]
We can write
\[ (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) \mathbf{k}= \mathbf{\nabla} \times \mathbf{a}\]
with \[\mathbf{a}\]
as defined above so the right hand side can be written \[\int \int_R ( \mathbf{\nabla} \times \mathbf{a}) \cdot \mathbf{k} \: dx \: dy \]
Green's Theorem can then be written
\[\oint_C \mathbf{a} \cdot d \mathbf{r} =\int \int_R ( \mathbf{\nabla} \times \mathbf{a}) \cdot \mathbf{k} \: dx \: dy \]
We can also write
\[\mathbf{a} \cdot d \mathbf{r} = \mathbf{a} \cdot \frac{d \mathbf{r}}{ds} ds = \mathbf{a} \cdot \mathbf{T} ds\]
where s is the distance along the curve. We can write
\[\mathbf{T} = \mathbf{k} \times \mathbf{n}\]
then\[\oint_C \mathbf{a} \cdot d \mathbf{r} = \oint_C \mathbf{a} \cdot \mathbf{T} ds = \oint_C \mathbf{a} \cdot ( \mathbf{k} \times \mathbf{n}) ds = \oint_C (\mathbf{a} \times \mathbf{k}) \cdot \mathbf{n}) ds\]
We can define a vector
\[\mathbf{b}= \mathbf{a} \times \mathbf{k}\]
.Then
\[\mathbf{b} = (P \mathbf{i} + Q \mathbf{j}) \times \mathbf{k} = Q \mathbf{i} - P \mathbf{j}\]
and \[\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \mathbf{\nabla} \cdot \mathbf{b}\]
The right hand side of Green's Theorem can then be written
\[\int \int_R \mathbf{\nabla} \cdot \mathbf{b} \: dx \: dy\]
.Green's Theorem can then be written
\[\oint_C (\mathbf{a} \times \mathbf{k}) \cdot \mathbf{n}) ds = \int \int_R \mathbf{\nabla} \cdot \mathbf{b} \: dx \: dy\]
. 
