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The changeofvariables formula with 3 (or more) variables is just like the formula for two variables. If we do a changeofvariables $\Phi$ from coordinates $(u,v,w)$ to coordinates $(x,y,z)$, then the Jacobian is the determinant $$\frac{\partial(x,y,z)}{\partial(u,v,w)} \ = \ \left  \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{matrix} \right ,$$ and the volume element is $$dV \ = \ dx\,dy\,dz \ = \ \left  \frac{\partial(x,y,z)}{\partial(u,v,w)}\right  du\,dv\,dw.$$ After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called cylindrical polar coordinates) and spherical coordinates (sometimes called spherical polar coordinates).
