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Microsoft infraredwindows 11

The change-of-variables formula with 3 (or more) variables is just like the formula for two variables. If we do a change-of-variables $\Phi$ 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).