General Triple Change of Variables

Conditions

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

Application

$$\begin{array}{rl}J& =\left|\det \left(\begin{array}{ccc}{x}_u^{\prime }& x_v^{\prime }& x_w^{\prime }\\ y_u^{\prime }& y_v^{\prime }& y_w^{\prime }\\ z_u^{\prime }& z_v^{\prime }& z_w^{\prime }\end{array}\right)\right|\\ 1/J& =\left|\det \left(\begin{array}{ccc}{u}_x^{\prime }& u_y^{\prime }& u_y^{\prime }\\ v_x^{\prime }& v_y^{\prime }& v_y^{\prime }\\ w_x^{\prime }& w_y^{\prime }& w_y^{\prime }\end{array}\right)\right|\end{array}$$