Verify step in proving Green's theorem (oriented 2-cells)

Question

One of the step in proving Green's theorem is:(from Advanced Calculus of Several Variables)
For every nice region that is the image of unit square $I^2$ under a suitable mapping.
The set $D$ subset in $R^2$ is called an oriented smooth 2-cell if there exists a one-to-one $C^1$ mapping $F:U \rightarrow R^2$ such that $F(I^2)=D$ and $I^2$ is a subset of open set $U$ and the Jacobian determinant of F is positive at each point of U.
So from Inverse function theorem we now that $F$ open sets send to open sets.
Otherwise, F will send point from $Int(I^2)$ to $Int(D)$.

But why F will send frontier of square $(I^2)$ to frontier of $D$ and keeping the positive orientation?

Answer 1

If $x$ is a frontier point of $I^2$ and $F(x)$ is not a frontier point of $D$, then there is an open set around $F(x)$ contained in $\mbox{Int } D$. The inverse image of that open set has to contain $x$, but is in $\mbox{Int } I^2,$ contradiction.

Since the Jacobian is positive, we know the orientation is preserved.