Define the cross product function $F: \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $F(v, w)=v \times w$.
The question is to prove that $F$ is differentiable and find its derivative.
Is the existence of Jacobian matrix enough to show that $F$ is differentiable?
If so, I will got the Jacobian matrix of $F$ by $ J_F(v,w) = \frac{\partial (F_1, F_2, F_3)}{\partial (v_1, v_2, v_3, w_1, w_2, w_3)} $
Am I on the right direction?