Alright, I'm clueless on how to kickstart this question.
The idea of the dirac delta function by itself is understandable, at least at my current level. But once the question starts throwing in things like charge or a sphere, I get really lost.
Here's the question:
$\int_{V}\vec{r}.\left ( \vec{d}-\vec{r} \right )\delta^{3}\left ( \vec{e}-\vec{r} \right )dV$ where $\vec{d}=\left \langle 1,2,3 \right \rangle ,\vec{e}=\left \langle 3,2,1 \right \rangle$, and V is a sphere of radius 1.5 centered at $\left ( 2,2,2 \right )$
I need to evaluate this integral.
$\delta^{3}\left ( \vec{e}-\vec{r} \right )$ tells me how much the dirac delta function is being shifted. In this case it is being shifted by $\vec{r}=\left \langle 2,2,2 \right \rangle$.
I can go no further. Help is required.
I think the integral with delta function means set a value, which is $$\int f(x)\delta (x-x_0)dx=f(x_0)$$ and in your case, it is $$\int \int \int _Vf(x,y,z)\delta (x-e_x)\delta (y-e_y)\delta (z-e_z)dxdydz$$ and is $\vec{e}\cdot (\vec{d}-\vec{e})$ when $\vec{e}$ in $V$ and $0$ when $\vec{e}$ is out of $V$. Hope it can help