Problem:
Find the jacobian of the vector valued function $\begin{pmatrix} f(x,y,z)\\ g(x,y,z) \\ h(x,y,z)\end{pmatrix}$ where $f(x,y,z) = \sqrt{x^2}$, $g(x,y,z) = \sqrt{\frac{z}{y}}$, and $h(x,y,z) = x^2-z$
Answer:
$\left(
\begin{array}{ccc}
 \frac{x}{\sqrt{x^2}} & 0 & 0 \\
 0 & -\frac{z}{2 y^2 \sqrt{\frac{z}{y}}} & \frac{1}{2 y \sqrt{\frac{z}{y}}} \\
 2 x & 0 & -1 \\
\end{array}
\right)$
{'source': 'AMPS', 'file': 'amps/mathematica/calculus/jacobian/30314.txt'}