# 3-D Dirac Delta Function

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1. Jan 31, 2017

### Dopplershift

1. The problem statement, all variables and given/known data

\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.
2. Relevant equations

3. The attempt at a solution

I know the following:

Suppose:

\int_V f(r) \delta^3(\vec{r}-\vec{r}') d\tau = f(\vec{r'})

(if r' is in the volume).

I'm just confused on how to plug in r' into f(r) which is

r^2 - 2\vec{r} \cdot \vec{r}'

Any help to get me started will be much appreciated.

2. Jan 31, 2017

### strangerep

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

3. Jan 31, 2017

### Dopplershift

Okay, so I can setup the integral.

\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

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

4. Jan 31, 2017

### Dopplershift

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