Showing a vector identity

Question

Problem: If $\phi$ and $\psi$ are smooth scalar fields, show that \begin{align*} \nabla \times (\phi \nabla \psi) = -\nabla \times (\psi \nabla \phi ) = \nabla \phi \times \nabla \psi .\end{align*}

Attempt at solution: First, I would use the vector identity \begin{align*} \nabla \times (\phi \mathbf{F}) = (\nabla \phi ) \times \mathbf{F} + \phi (\nabla \times \mathbf{F}). \end{align*} In our case $\nabla \psi = \mathbf{F}$. Hence we have that \begin{align*} \nabla \times (\phi \nabla \psi) = (\nabla \phi) \times (\nabla \psi) + \phi(\nabla \times \nabla \psi). \end{align*} Then I'm not really sure what to do next. Can I conclude on the RHS that $\phi(\nabla \times \nabla \psi) = 0$, because it is a curl grad? If that's the case, I still don't know how to arrive at the other expressions.

Any help please?

Answer 1

Yes that's zero, since it's a curl of grad. For the other expansion, try to expand $0=\nabla\times\nabla(\phi\psi)$.

Answer 2

Just to show you the ease of index notation, lookie here:

$$[\nabla \times (\phi\nabla\psi)]_p = \epsilon_{pqr}\partial_q(\phi\partial_r\psi) = \epsilon_{pqr}(\partial_q\phi)\partial_r\psi + \epsilon_{pqr}\phi\partial_q\partial_r\psi \\ = [(\nabla \phi)\times(\nabla \psi)]_p + [\phi(\nabla\times(\nabla \psi))]_p$$

Then showing that $\phi(\nabla\times(\nabla \psi))=0$ is easy too. I don't want to carry around the $\phi$, though, so convince yourself that if $\nabla\times(\nabla \psi)=0$, then $\phi(\nabla\times(\nabla \psi))=0$. Now, for the proof:

$$\begin{array}\ [\nabla\times(\nabla \psi)]_p = \epsilon_{pqr}\partial_q\partial_r\psi \\ = \epsilon_{pqr}\partial_r\partial_q\psi & \text{Clairaut's theorem} \\ =\epsilon_{prq}\partial_q\partial_r\psi & \text{relabel dummy indices: $q \leftrightarrow r$} \\ = -\epsilon_{pqr}\partial_q\partial_r\psi & \epsilon_{pqr} = -\epsilon_{prq}\end{array}$$

And from here we can see that $[\nabla\times(\nabla \psi)]_p=0$ because it's equal to both $\epsilon_{pqr}\partial_q\partial_r\psi$ AND $-\epsilon_{pqr}\partial_q\partial_r\psi$ and the only scalar field identically equal to its negative is the $0$ field.

Therefore $\nabla \times (\phi\nabla\psi) = (\nabla \phi)\times(\nabla \psi)$.$\ \ \ \ \ \square$