When is the Jacobian determinant positive

Question

Suppose we have $F(u, v) = (x, y)$.

The Jacobian of this mapping is denoted by $\dfrac{\partial (x,y)}{\partial (u,v)}$.

Is the Jacobian a positive real number when I evaluate at point $(u, v)$?

I calculate Jacobian using $ \dfrac{\partial (x,y)}{\partial (u,v)} = \det \begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{bmatrix} = \det \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = 1 - 0=1$.

Seems like the Jacobian determinant is always a positive real number under this mapping.

I don't know if I am doing the right thing.

Answer 1

The Jacobian is not a real number. For a map between $\mathbb R^n$ to $\mathbb R^m$ it is at each point where it is evaluated a matrix. In your case a square matrix of dimension$2$.

And the determinant of the Jacobian is a real number. Not a positive one in general.

For your specific map, the determinant is equal to $1$.

Answer 2

The Jacobian determinant $|J|$ at some point $P$ is positive if the orientation at that point is preserved by the map. Further, a negative Jacobian determinant at some point implies that orientation is reversed there.