Curl of a Curl of a Vector field Question

Question

Question: For the given vector function:

$$\vec{F}(x,y,z)=xz\vec{i}+y\vec{j}+y^2\vec{k}$$

Compute the expression:

$$ (\delta _{3l}\delta _{jm}-\delta _{3m}\delta _{jl}) \frac{\partial^2F_m}{\partial x_j\partial x_l} $$

at the point P=(1,0,1)

I understand for a vector field $\textbf{F}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{F}\right)=\nabla\left(\nabla\cdot\textbf{F}\right)-\nabla^2\textbf{F}$$ where $\nabla$ is the usual del operator and $\nabla^2$ is the vector Laplacian.

I worked out so far that $ (\delta _{3l}\delta _{jm}-\delta _{3m}\delta _{jl}) $ is equal too $\varepsilon _{i3j}\varepsilon _{ilm}$

But i'm confused on how to go from here, if anyone can help - would be greatly appreciated.

Answer 1

What I have so far is:

We look at the number next to the $\delta $, we then just have to work out the curl by doing the determinant matrix of our $\vec{F}$ and our partial derivatives of ${x,y,z}$, we get an answer for this in this the form ${a\vec{i},b\vec{j},c\vec{k}}$. Then we work out the curl of this equation ${a\vec{i},b\vec{j},c\vec{k}}$. We then plug our point in that corresponds to the $\delta$ value.