What does this notation mean: $\cfrac{\partial (\Psi,\xi)}{\partial (x,y)}$

Question

What does this notation mean: $\cfrac{\partial (\Psi,\xi)}{\partial (x,y)}$ ?

It's from an old paper published in 1953. It could be an old notation that is not used anymore, but I am not sure what it means.

Context:

Reference: Kawaguti, M. (1953). Numerical solution of the Navier-Stokes equations for the flow around a circular cylinder at Reynolds number 40. Journal of the Physical Society of Japan, 8(6), 747-757.

Answer 1

In modern usage it typically denotes the Jacobian matrix of the mapping $(x,y)\mapsto (\Psi,\zeta).$ In this case, it is being used to denote the determinant of this matrix, i.e. $$\cfrac{\partial (\Psi,\xi)}{\partial (x,y)}=\frac{\partial \Psi}{\partial x}\frac{\partial \xi}{\partial y}-\frac{\partial \Psi}{\partial y}\frac{\partial \xi}{\partial x}.$$ See this Wikipedia section, where you can find a very similar equation for time-dependent flow.

Answer 2

This is a notation for the determinant of partial derivatives. $$\cfrac{\partial(\Psi,\xi)}{\partial (x,y)}= $$

$$ det \begin {bmatrix} \frac{\partial \Psi}{ \partial x} &\frac{\partial \Psi}{ \partial y} \\ \frac{\partial \xi}{\partial x} &\frac{\partial \xi}{\partial y}\end{bmatrix} $$