3-D Dirac Delta Function

  • #1

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.

Homework Equations




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.
 

Answers and Replies

  • #2
strangerep
Science Advisor
3,273
1,229
$$\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau $$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.
 
  • #3
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}##)
 
  • #4
Never Mind, I figured it out. I was overthinking the problem. Thanks for your help! :)
 

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