To find the partial derivatives of the functions $u$, $v$, and $w$ with respect to the variables $x$, $y$, and $z$, we need to compute the Jacobian matrix. The Jacobian matrix $J$ of the transformation from $(x, y, z)$ to $(u, v, w)$ is given by:

[ J = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \