- #1

TheFerruccio

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**Problem Statement**

Prove the following identity is true, using indicial notation:

[itex]\nabla\times(\nabla \vec{v})^T = \nabla(\nabla\times\vec{v}) [/itex]

**Attempt at Solution**

Let [itex]U_{kp} = v_{k,p}[/itex]

Then LHS:

[itex]\nabla\times(U_{kp})^T = \varepsilon_{ijk}U_{pk,j}[/itex]

RHS:

[itex]\nabla(\varepsilon_{ijk}v_{k,j})=\nabla(\varepsilon_{ijk}U_{kj})=\varepsilon_{ijk}U_{kj,p}[/itex]

I understand that I can swap dummy indices around however I like, and simply rename free indices, but, even then, RHS does not equal LHS, and I do not know what I am doing wrong. This is the final result I get.

[itex]\varepsilon_{ijk}U_{pk,j}=\varepsilon_{ijk}U_{kj,p}[/itex]

I know that I am not supposed to change both sides, but I was doing this for reference, to get an idea of where I was going. I was going to reconstruct to end up with the RHS. What am I doing wrong?