Define the Jacobian, $J$, of the one-to-one transformation

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

Give a careful explanation of the result

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

where

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

and the region $D$ maps under the transformation to the region $\Delta$.

Consider the region $D$ defined by

$$x^{2}+\frac{y^{2}}{k^{2}}+z^{2} \leqslant 1$$

and

$$\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{k^{2} \alpha^{2}}-\frac{z^{2}}{\gamma^{2}} \geqslant 1$$

where $\alpha, \gamma$ and $k$ are positive constants.

Let $\tilde{D}$ be the intersection of $D$ with the plane $y=0$. Write down the conditions for $\tilde{D}$ to be non-empty. Sketch the geometry of $\tilde{D}$ in $x>0$, clearly specifying the curves that define its boundaries and points that correspond to minimum and maximum values of $x$ and of $z$ on the boundaries.

Use a suitable change of variables to evaluate the volume of the region $D$, clearly explaining the steps in your calculation.