Write down an expression for the Jacobian $J$ of a transformation

$$(x, y, z) \rightarrow(u, v, w)$$

Use it to show that

$$\int_{D} f d x d y d z=\int_{\Delta} \phi|J| d u d v d w$$

where $D$ is mapped one-to-one onto $\Delta$, and

$$\phi(u, v, w)=f(x(u, v, w), y(u, v, w), z(u, v, w))$$

Find a transformation that maps the ellipsoid $D$,

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leqslant 1$$

onto a sphere. Hence evaluate

$$\int_{D} x^{2} d x d y d z$$