1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integral with divergence thm

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data
    Evaluate the integral
    [tex]\int\limits_{V=\infty} e^{-r} \left[ \nabla \cdot \frac {\widehat{r}} {r^2} \right] , d^3 x[/tex]

    2. Relevant equations
    Divergence theorem:
    [tex]\int\limits_{V} \left ( \nabla \cdot A \right ) \, d^3 x
    = \oint\limits_{S} A \cdot \, da}

    3. The attempt at a solution
    I know that I have to apply the div theorem somewhere, but this [tex]e^{-r}[/tex] is confusing and what does it mean if the lower limit V is infinity?
    I haven't seen the integral of [tex]\frac{1}{e^r} [/tex] before but I'm kinda guessing
    [tex] \int \frac{1}{e^r} \, dr
    = \frac{1}{e^r} \int \frac{1}{u} \frac{du}{e^r}
    = ln(e^r)
    = r
    where I used a substitution [tex]u=e^r[/tex] and [tex]du= e^r dr[/tex]
    Last edited: Aug 14, 2010
  2. jcsd
  3. Aug 15, 2010 #2


    User Avatar
    Homework Helper

    What is the divergence of vec(r)/r^2?

  4. Aug 17, 2010 #3
    It's defined as
    [tex]4 \pi \delta x \delta y \delta z [/tex]

    but then I don't know how to apply Stokes' (which I guess to use because of the [tex]d^3 x [/tex] and V in the integral. Could I split it into a triple integral and [tex]\delta x dx [/tex] at a time?
  5. Aug 17, 2010 #4

    Char. Limit

    User Avatar
    Gold Member

    This is wrong.

    [tex]\frac{1}{e^r} = e^{-r}[/tex]

    [tex]\int e^{-r} dr[/tex]

    [tex]u=-r, du=-dr, -du=dr[/tex]

    [tex]\int -e^u du = -e^u = -e^{-r} = \frac{-1}{e^r}[/tex]

    The integral of e^-r isn't r, as that would imply that e^-r is a constant number.
  6. Aug 17, 2010 #5
    I agree, my above reasoning was useless

    Ok so I can integrate the [tex]e^-r[/tex] but I don't think that really matters when there's a delta in the integral... my main problem is how to solve a third order delta integral, probably using the Divergence theorem because of the third order and volume. So we have

    [tex]4 pi \int_{V=\infty} e^{-r} {\delta}^3 x z d^3 x[/tex]
  7. Aug 19, 2010 #6


    User Avatar
    Homework Helper

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook