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3-D Dirac Delta Function

  1. Jan 31, 2017 #1
    1. The problem statement, all variables and given/known data
    \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.
    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Jan 31, 2017 #2

    strangerep

    User Avatar
    Science Advisor

    First, write out your integral as an explicit triple integral.
     
  4. Jan 31, 2017 #3
    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}##)
     
  5. Jan 31, 2017 #4
    Never Mind, I figured it out. I was overthinking the problem. Thanks for your help! :)
     
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