Proof of Jacobian Factor for Transformation of Area Integral
\[(x,y)=(x(u,v), y(u,v))\]
then\[\int \int_R \: dx \: dy = \int \int_R \frac{ \partial (x,y)}{\partial (u,v)} \: du \: dv\]
where\[\frac{ \partial (x,y)}{\partial (u,v)} = \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{array} \right| = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\]
From Green's Theorem
\[\frac{1}{2} \oint_C x \: dy - y \: dx = \int \int_R dx \: dy \]
Concentrate on the left hand side.
\[dx = \frac{\partial x}{\partial u} du + \frac{\partial x}{\partial v} dv, \: dy = \frac{\partial y}{\partial u} du + \frac{\partial y}{\partial v} dv\]
\[\begin{equation} \begin{aligned} A &= \frac{1}{2} \oint_C x \: dy - y \: dx \\ &= \frac{1}{2} \oint_C x (\frac{\partial y}{\partial u} du + \frac{\partial y}{\partial v} dv)- y (\frac{\partial x}{\partial u} du + \frac{\partial x}{\partial v} dv) \\ &= \frac{1}{2} \oint_C (x \frac{\partial y}{\partial u} - y \frac{\partial x}{\partial u}) du + (x \frac{\partial y}{\partial v}- y \frac{\partial x}{\partial v}) dv \\ &= \frac{1}{2} \int \int_R \frac{\partial }{\partial u}(x \frac{\partial y}{\partial v}- y \frac{\partial x}{\partial v}) - \frac{\partial}{\partial v}(x \frac{\partial y}{\partial u} - y \frac{\partial x}{\partial u}) \: du \: dv \\ &= \frac{1}{2} \int \int_R ( \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} +x \frac{\partial^2 y}{\partial u \partial v} - \frac{\partial y}{\partial u} \frac{\partial x}{\partial v} -y \frac{\partial^2 x}{\partial u \partial v} \\ &- \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} - x \frac{\partial^2 y}{\partial v \partial u} + \frac{\partial y}{\partial v} \frac{\partial x}{\partial u} + y \frac{\partial^2 x}{\partial v \partial u}) du \: dv \\ &= \int \int_R ( \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}) du \: dv \\ &= \int \int_R \frac{\partial (x,y)}{\partial (u,v)} \: du \: dv \end{aligned} \end{equation}\]
