How about setting
I think you would have
because you essentially have the Laplacian in two dimensions. If you ignore the z dimension, you can just copy down the version for cylindrical coordinates, which is what I did.
See here for a derivation. You'll notice that they've multiplied out the product rule that I have not multiplied out.
Please provide a reference for the equation
Like I said before, I think this equation is incorrect. The units don't work out. For the first term, the and units cancel out, because they're both length. That leaves you with units of divided by units of length. On the second term, the and units cancel out, as before, but the units of are NOT units of divided by units of length. Therefore, you can't add the terms because they don't have the same units.
Actually, they don't have that. What they actually have is this:On the site, they have
...
Operators are very slippery creatures. There's a HUGE difference between
and
Think of the first as an operator: it's going to look to its right and try to find something to operate on (the operation in question is partial differentiation with respect to phi). On the other hand, the second is not an operator, because the differentiation has already taken place. It's just a function.
Does that make sense?