I can't fathom the step I have highlighted in green. Am I using the chain rule in 3 dimensions? What is it that I am transforming here?
2026-04-03 15:29:50.1775230190
Transformation rule for partial derivatives
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Yes, you are using the chain rule. In cylindrical coordinates $r=r(x,y)=\sqrt{x^2+y^2}$. So, as $r$ only has a dependance on $x$ and $y$:
$$\frac{\partial \phi}{\partial r}=\frac{\partial x}{\partial r}\frac{\partial \phi}{\partial x}+\frac{\partial y}{\partial r}\frac{\partial \phi}{\partial y}$$