Derivative\Jacobian Matrix of $f:\mathbb{R}^3\times\mathbb{R}^3\to \mathbb{R}^3$

Question

I am working with the derivative of the cross-product of two vectors, and am trying to understand what the Jacobian matrix of derivatives would look like. In particular, I am confused about the size.

We have $f:\mathbb{R}^3\times\mathbb{R}^3\to \mathbb{R}^3$ as the cross product function. Would the Jacobian matrix of $Df(x,y,z)$ be a $3\times 3$ or a $3\times 6$ sized matrix? I guess this boils down to how I interpret $\mathbb{R}^3\times\mathbb{R}^3$.

Thank you for your input.

Answer 1

It's a $3\times 6$ matrix. I don't know how you can interpret $\Bbb R^3\times\Bbb R^3$ and decide it's just $\Bbb R^3$. :)