- #1
pellman
- 683
- 6
Claim:
\nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x})
Anyone know of a proof of this? (or a reference which covers it?) We need to show that
\frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0).
The claimed identity can be seen in the solution for the electric field of a point charge in EM theory, where
\vec{E}=\frac{q}{r^2}\hat{e}_r
is the solution to
\nabla \cdot \vec{E}=4\pi q\delta^3(\vec{x})
It is easy to show in this case that \nabla \cdot \vec{E}=0 everywhere but the origin, but I don't know how to show that the delta function relation holds at the origin.
\nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x})
Anyone know of a proof of this? (or a reference which covers it?) We need to show that
\frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0).
The claimed identity can be seen in the solution for the electric field of a point charge in EM theory, where
\vec{E}=\frac{q}{r^2}\hat{e}_r
is the solution to
\nabla \cdot \vec{E}=4\pi q\delta^3(\vec{x})
It is easy to show in this case that \nabla \cdot \vec{E}=0 everywhere but the origin, but I don't know how to show that the delta function relation holds at the origin.
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