How in general can we find a functional relationship between two functions
$$ u(x,y) ~, v(x,y)~$$
when Jacobian
$$\begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}$$
vanishes? i.e., what pde/ode can be used to find the dependence or link equation?
How does it change for three functions and three variables?
Appreciate all help.