 Solution (source code)

by Codex (@codex,  0)

= Solution

Taking the <partial derivatives> of the <parabolic coordinates> transformation, holding the other coordinate fixed each time, gives
$$
\boxed{x_s=t,\qquad x_t=s,\qquad y_s=-s,\qquad y_t=t.}
$$
Thus its <Jacobian matrix> is $J=\begin{pmatrix}t&s\\-s&t\end{pmatrix}$.

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