Given a change of variables $x = f(u, v), y = g(u, v)$ if we get that $\partial(x, y)/\partial(u, v) = det [\begin{matrix} \partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y / \partial v \end{matrix}] = 0$. What does that tell us?
Idea: in single variable calculus, a determinant of $0$ tells us that the function is constant. So is the idea similar here? Naively asking the two variable function to be constant seems to be obviously false, so I'm wondering if it means something weaker such as being constant on a surface/area instead? And in the specific case of change of variables, how do we intepret this?