Triple integrals

If we want to change a triple integral in xyz to a triple integral in another coordinate system, say uvw where the change of coordinates is given by

$$x = x(u,v,w), \quad y = y(u,v,w), \quad z = z(u,v,w).$$

we need to compute what dV is in the new coordinate system. To find this volume element we carry out the three dimensional version finding the area element and use the fact that if $\vec{a}, \vec{b}, \vec{c}$ are the vectors along three edges of a parallelepiped, then the volume is $\vec{a} \cdot (\vec{b} \times \vec{c})$. This will eventually lead us to a three dimensional version of the Jacobian:

$$J = \frac{\partial (x,y,z)}{\partial (u,v,w)} = \left\vert \... ...tial w}& \frac{\partial z}{\partial w} \end{array} \right\vert.$$

Now,

$$dV = \left\vert \frac{\partial (x,y,z)}{\partial (u,v,w)} \right\vert du dv dw.$$

It is a very good exercise to verify that the Jacobian for cylindrical coordinates is r and the Jacobian for spherical is $\rho^2 \sin \phi$.