General Double Change of Variables

Conditions

• For the transform $T:P^{\prime }(u,v)\to P(x,y)$
• Given
  • $x(u,v)$ and $y(u,v)$ have continuous partial derivatives
  • $T$ is one-to-one (except possibly on the boundaries) → $T^{-1}$ exists

Application

$${\iint }_{D}f(x,y)dA={\iint }_{D^{\prime }}f(x(u,v),y(u,v))J(u,v)d{A}^{\prime }$$ $$J=\left|\det \left(\begin{array}{cc}{x}_u^{\prime }& x_v^{\prime }\\ y_u^{\prime }& y_v^{\prime }\end{array}\right)\right|=\left|1/\det \left(\begin{array}{cc}{u}_x^{\prime }& v_x^{\prime }\\ u_y^{\prime }& v_y^{\prime }\end{array}\right)\right|$$

The Jacobean ($J$) means how much the area of a partition changes with the transformation