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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?

Rita
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  • The existence of the Jacobian matrix, i.e. of all partial derivatives, is not sufficient. But if (like here) they are continuous functions of $(v,w)$ then fine: $F$ will even be $C^1$ i.e. continuously differentiable. Note that here, the $F_i$'s are polynomial in the $v_j,w_k$'s, hence $C^\infty.$ You can also argue that your $F$ is bilinear (hence continuous since the dimension is finite), so $DF_{(u,v)}(h,k)=F(u,k)+F(h,v).$ – Anne Bauval Apr 03 '23 at 10:01
  • Does this answer your question? Derivative of cross product or one of the many other related posts? – Anne Bauval Apr 03 '23 at 10:12
  • Your title mentions high order derivatives but your body does not. – Anne Bauval Apr 03 '23 at 10:13

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