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In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere. I've tried Googling for hours now, and I've legitimately come up with no example with a good accompanying explanation, of how this particular identity is working. This is the identity in question. Any help would be appreciated.

##

(\vec{r}\times\nabla)\psi =

\nabla\times \vec{r}\psi - \psi(\nabla\times \vec{r})

##

EDIT: I've also seen this identity on Wikipedia, which _may_ satisfy my requirement if I'm thinking of the curl operator right.##\nabla \times(\psi \vec{r}) = \psi(\nabla \times \vec{r}) + \nabla\psi\times\vec{r}##

(\vec{r}\times\nabla)\psi =

\nabla\times \vec{r}\psi - \psi(\nabla\times \vec{r})

##

EDIT: I've also seen this identity on Wikipedia, which _may_ satisfy my requirement if I'm thinking of the curl operator right.

For the first term on the RHS, we can use the anticommutativity of the cross product, and state

##\psi(\nabla \times \vec{r}) = - \psi (\vec{r} \times \nabla)##and then the identity becomes a trivial moving around of terms. Am I thinking about that in the right way?

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