|| $\frac{\partial F}{\partial x} \times \frac{\partial F}{\partial y}$||= ||$J_x F$||

Question

Let $U\subseteq \mathbb{R}^2$ and $ F:U\rightarrow \mathbb{R}^3$, $F =(F_1,F_2,F_3) \in C^1(U)$. Show that the norm of the vector $\frac{\partial F}{\partial x} \times \frac{\partial F}{\partial y}$ is equal to the norm of the vector ($\frac{\partial F_1}{\partial x}, \frac{\partial F_2}{\partial x},\frac{\partial F_3}{\partial x})$. I need to know some general result?

Answer 1

Is false. If $F = (x,y,xy)$, $$\frac{\partial F}{\partial x}\times\frac{\partial F}{\partial y} = (1,0,y)\times(0,1,x) = (-y-x,1),$$ but $$\|(-y-x,1)\|\ne\|(1,0,y)\|.$$