Jacobian matrix of $F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ Compute a matrix representative of $F_*$, where $F$ is
$$F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$$
$$(u,v)\to u\times v$$
and $u\times v$ is the vector cross product of $u$ and $v$.
We can take the charts $(U,\phi)=(\mathbb{R}^3 \times \mathbb{R}^3, Id_{\mathbb{R}^3 } \times Id_{\mathbb{R}^3})$ and $(V,\psi)=(\mathbb{R}^3, Id_{\mathbb{R}^3})$. And then we have
$[F_*]=$ Jacobian matrix.
But I don't know how to compute the Jacobian matrix of this map.
Attempt:
Since $(u^1,u^2,u^3)\times (v^1,v^2,v^3)=(u^2 v^3-u^3v^2,u^1 v^3-u^3v^2, u^1 v^2-u^2 v^1)$
Our matrix would be
\begin{bmatrix}
0 & v^3 & -v^2 & 0 & -u^3 & u^2\\
-v^3 & 0 & v^1 & u^3 & 0 & -u^1 \\
v^2 & -v^1 & 0 & -u^2 & u^1 & 0
\end{bmatrix}
Any help would be appreciated. Thanks
 A: Since you're dealing with the identity chart, this becomes a simple multivariable calculus problem, so you can forget all about manifolds. $F$ is a bilinear map so the derivative is very simple. Bilinear maps should be thought of as products, so we're just going to apply the product rule here: given a point $(u,v)\in\Bbb{R}^3\times \Bbb{R}^3$ and $(\xi,\eta)\in\Bbb{R}^3\times \Bbb{R}^3$, we have
\begin{align}
DF_{(u,v)}[\xi,\eta]&= F(u,\eta)+F(\xi,v)\\
&=u \times \eta + \xi\times v 
\end{align}
(The really short way of summarizing all of this is to say $d(u\times v)=u\times dv+ du\times v$). This is now a linear function of $(\xi,\eta)$, so finding the matrix representation means we successively plug in basis vectors. For example, calculate the following six quantities:

*

*$DF_{(u,v)}[(1,0,0,0,0,0)] = u\times (1,0,0)+(0,0,0)\times v = u\times (1,0,0)=(0,u_3,-u_2)$

*$DF_{(u,v)}[(0,1,0,0,0,0)]$

*$DF_{(u,v)}[(0,0,1,0,0,0)]$

*$DF_{(u,v)}[(0,0,0,1,0,0)]$

*$DF_{(u,v)}[(0,0,0,0,1,0)]$

*$DF_{(u,v)}[(0,0,0,0,0,1)]$
I leave the other computations to you. For each of these you should get certain elements of $\Bbb{R}^3$ as I've demonstrated for the first one. Now, take these $6$ elements of $\Bbb{R}^3$ and arrange them in columns to form a $3\times 6$ matrix. That is the matrix representation of $DF_{(u,v)}$ in terms of standard bases, and equivalently it is also the matrix representation of $F_{*,(u,v)}$ in terms of the corresponding bases.
