Green's Theorem in Vector Form

Green's Theorem stats that if
$P,Q$
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
$\oint_C (\mathbf{a} \times \mathbf{k}) \cdot \mathbf{n}) ds = \int \int_R \mathbf{\nabla} \cdot \mathbf{b} \: dx \: dy$
.