When Is the Jacobian Zero?

For sufficiently smooth maps (continuously differentiable is enough), the Jacobian is identically zero if and only if the image has area zero.

What does it mean when the Jacobian is zero?

If the Jacobian is zero, it means that there is no change whatsoever, and this means you get an overall change of zero at that point (with respect to the rate of change with respect to the expansion and contraction with respect to the entire volume).

What if Jacobian is not zero?

Inverse. Conversely, if the Jacobian determinant is not zero at a point, then the function is locally invertible near this point, that is, there is a neighbourhood of this point in which the function is invertible.