Substitution Theorem

Suppose $\vec{T}(u,v)$ is a 1-1, Differentiable Transformation that maps the Region $G$ in the $uv$ plane to the region $R$ in the $xy$ plane. Then:

$${\iint }_{R}f(x,y)dxdy={\iint }_{G}f(\vec{T}(u,v))\det \left(J_{\vec{T}}\right)dudv$$

where $\det J_{\vec{T}}$ is the Jacobian Determinant of $\vec{T}(u,v)$