# 3-D Dirac Delta Function

## Homework Statement

\begin{equation}
\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau
\end{equation}

where:
\begin{equation}
\vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z}
\end{equation}

Where d $\tau$ is the volume element, and V is a solid sphere with radius 4, centered at the origin.

## The Attempt at a Solution

I know the following:

Suppose:
\begin{equation}
\int_V f(r) \delta^3(\vec{r}-\vec{r}') d\tau = f(\vec{r'})
\end{equation}
(if r' is in the volume).

I'm just confused on how to plug in r' into f(r) which is
\begin{equation}
r^2 - 2\vec{r} \cdot \vec{r}'
\end{equation}

Any help to get me started will be much appreciated.

strangerep
$$\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau$$where:
$$\vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z}$$ where ##d\tau## is the volume element, and V is a solid sphere with radius 4, centered at the origin.
First, write out your integral as an explicit triple integral.

First, write out your integral as an explicit triple integral.

Okay, so I can setup the integral.

\begin{equation}
\int_v (r^2 - 2\vec{r} \cdot \vec{r}') \delta_x(x-x_0) \delta_y (y-y_0) \delta_z (z-z_0) dx dy dz
\end{equation}

I guess I'm confused how I plug ##\vec{r}'## into f(##\vec{r}##)

Never Mind, I figured it out. I was overthinking the problem. Thanks for your help! :)