what does it mean by determinant of Jacobian matrix = 0?

Question

I have an example: $$ u={x+y\over 1-xy} $$ $$ v = \tan^{-1}(x)+\tan^{-1}(y) $$ So by calculating the determinant of the Jacobian matrix I get zero. Does it mean there is no functional relationship between u and v? What does $|J|=0$ mean?

Answer 1

In this case, there is a functional relationship between $u$ and $v$: in fact $u = \tan(v)$. Thus the transformation $(x,y) \to (u,v)$ is not invertible: there is no way to get $x$ and $y$ back from $u$ and $v$.