Change of Variables for uv-Plane Transformation

Theorem

• If $D,D^{\ast }$ are Elementary Region in plane
• If $T:D^{\ast }\to D$ is $C^1$ Class Function
• If $T$ is Injective Then, for any integrable region $f:D\to \mathbb{R}$, we have: $\int {\int }_{D}f(x,y)dxdy=\int {\int }_{D^{\ast }}f(x(u,v),y(u,v))|\frac{\partial (x,y)}{\partial (u,v)}|dudv$