Jacobian matrix of $F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$

Question

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

Answer 1

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.