# Help with tensor notation and curl

1. Oct 3, 2011

### stylophora

1. The problem statement, all variables and given/known data

Show that $\nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla a$

2. Relevant equations

$$\nabla \times (\nabla \phi) = 0$$
$$\nabla \cdot (\nabla \times a) = 0$$

3. The attempt at a solution

I started with breaking the LHS into two components. Observing that:
$$a\times(\nabla\times a) = 0.5 \nabla(a\cdot a) - a\cdot\nabla a$$

Taking the cross product of both sides:
$$\nabla\times a\times(\nabla\times a) = 0.5\nabla\times\nabla(a\cdot a) - \nabla\times(a\cdot \nabla a)$$

Recognizing that the term on the right corresponds to our initial equation.
$$\nabla \times (a \cdot \nabla a) = -\nabla\times a\times(\nabla\times a)$$

Unfortunately, I am sort of stuck here. One way that I have thought to go about it is by calling the LHS:
$$-\nabla\times a\times(\nabla\times a) = -\nabla\times c$$
where c = a\times(\nabla\times a)
I am confused on how to expand the above out using levi civita. I know that:
$$(\nabla \times c)_i = \epsilon_{ijk} \frac{dc_k}{dx_j}$$
But substituting in for c isn't making sense to me.

Sorry for the very rough attempt at a solution. I only started doing vector calc a week ago.

Last edited: Oct 3, 2011