## Interchanging Functions and Coordinates in Green's Theorem

Green's Theorem states that for differentiable functions
$P(x,y), \: Q(x,y)$
defined on a region
$R$
with boundary
$C$
;
$\oint_C P \; dx + Q \: dy = int \int_R (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) \: dx \: dy$

Interchanging
$P,Q$
and
$x,y$
respectively gives
$\oint_C Q \; dy + P \: dx = int \int_R (\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x} ) \: dx \: dy$

The Left hand side is the same as before, but the right hand side is reversed in sign, which amounts to reversing the orientation of the curve.
With
$\mathbf{a} = P \mathbf{i} + Q \mathbf{j}$
and
$\mathbf{r}= x \mathbf{i} +y \mathbf{j} \rightarrow d \mathbf{r}= dx \mathbf{i} +dy \mathbf{j}$

$\oint_C \mathbf{r} \cdot d \mathbf{r} = - \oint_{-C} \mathbf{r} \cdot d \mathbf{r}$