Geen's Theorem in Polar Coordinates

With  
\[x= r \: cos \theta\]
,  
\[dx= cos \: \theta dr - r \: sin \: \theta d \theta\]

With  
\[y= r \: sin \theta\]
,  
\[dy= sin \: \theta dr + r \: cos \: \theta d \theta\]

Then  
\[\frac{1}{2} \oint x \: dy - y \: dx = \frac{1}{2} \int r \; cos \; \theta (sin \: \theta dr + r \: cos \: \theta d \theta) -r \: sin \: \theta (cos \: \theta dr - r \: sin \: \theta d \theta) = \frac{1}{2} \oint r^2 \: d \theta\]
 

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