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$\nabla \cdot \vec E(r) = \dfrac {1}{r^2} \dfrac {d(r^2 E)}{dr} \ne \dfrac{dE}{dr}$ in spherical coordinates.

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The differential and integral forms should in principle always lead to the same result, since they are related to each other via Gauss's theorem. (e.g. see What are the differences between the differential and integral forms of (e.g. Maxwell's) equations? ). In this case you have not applied the differential form correctly, because you have used an ...

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You are confusing chemistry and physics. In chemistry you may have learnt that when metals react, they give up electrons (the are oxidized). That is because there are usually a small number of electrons in the outermost orbit of the atom, and when these are released the atom is left with a very stable electron configuration. But when you are looking at a ...

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