(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that [itex]\nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla a[/itex]

2. Relevant equations

[tex]\nabla \times (\nabla \phi) = 0[/tex]

[tex]\nabla \cdot (\nabla \times a) = 0[/tex]

3. The attempt at a solution

I started with breaking the LHS into two components. Observing that:

[tex]a\times(\nabla\times a) = 0.5 \nabla(a\cdot a) - a\cdot\nabla a[/tex]

Taking the cross product of both sides:

[tex]\nabla\times a\times(\nabla\times a) = 0.5\nabla\times\nabla(a\cdot a) - \nabla\times(a\cdot \nabla a)[/tex]

Recognizing that the term on the right corresponds to our initial equation.

[tex]\nabla \times (a \cdot \nabla a) = -\nabla\times a\times(\nabla\times a)[/tex]

Unfortunately, I am sort of stuck here. One way that I have thought to go about it is by calling the LHS:

[tex]-\nabla\times a\times(\nabla\times a) = -\nabla\times c[/tex]

where c = a\times(\nabla\times a)

I am confused on how to expand the above out using levi civita. I know that:

[tex](\nabla \times c)_i = \epsilon_{ijk} \frac{dc_k}{dx_j} [/tex]

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Help with tensor notation and curl

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**