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  
\[\mathbf{r}= x \mathbf{i} +y \mathbf{j} \rightarrow d \mathbf{r}= dx \mathbf{i} +dy \mathbf{j}\]
 
\[\oint_C \mathbf{r} \cdot d \mathbf{r} = - \oint_{-C} \mathbf{r} \cdot d \mathbf{r}\]
 

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