Wikipedia has a nice writeup on hyperbolic coordinates: https://en.wikipedia.org/wiki/Hyperbolic_coordinates
In particular, they introduce the parametrization \begin{equation} x = ve^u \\ y = ve^{-u}. \end{equation} I would like to find the area element, $dA$, in this coordinate system. We have the Jacobian: \begin{pmatrix} ve^u & e^u \\ -ve^{-u} & e^{-u} \end{pmatrix} which has determinant $2v$. Therefore, $dA = 2|v|dudv$. Is this correct? This is quite an interesting result if so. Much simpler than I expected, and very similar to the polar area element $rdrd\theta$.
edit: $v$ $\rightarrow$ $|v|$
Yes it is correct. Hyperbolic “rectangles” get bigger and bigger as you move away from the origin, same effect than moving away from the pole in polar coordinates.