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!

Gauss' Theorem - Divergence Theorem for Sphere

  1. Oct 21, 2015 #1
    1. The problem statement, all variables and given/known data
    Using the fact that [itex] \nabla \cdot r^3 \vec{r} = 6 r^2[/itex] (where [itex]\vec{F(\vec{r})} = r^3 \vec{r}[/itex]) where S is the surface of a sphere of radius R centred at the origin.

    2. Relevant equations
    [tex]
    \int \int \int_V \nabla \cdot \vec{F} dV =\int \int_S \vec{F} \cdot d \vec{S}
    [/tex]
    That is meant to be a double surface integral but not sure how to do that in latex

    3. The attempt at a solution
    Very new to this, so be kind :)

    for the LHS
    [tex]
    \int \int \int_V \nabla \cdot \vec{F} dV = 6 \int \int \int_V r^2 dV \\
    = 6 \int_0^R r^2 r^2 dr \int_0^{2 \pi} d\phi \int_0^{\pi} \sin{\theta} d\theta \\
    = 6 [\frac{1}{5}r^5]_0^R [\phi]_0^{\pi} [\cos{\theta}]_0^{2 \pi} \\
    = 6(\frac{1}{5} R^5) (2 \pi ) (2) = \frac{24}{5} \pi R^5
    [/tex]

    for the RHS
    [tex]
    \int \int_S \vec{F} \cdot d\vec{S} = \int \int_S r^3 \vec{r} \cdot d \vec{S} = \int \int_S r^3 \vec{r} \cdot \vec{\hat{n}} dS \\
    [/tex]

    Then I calc [itex]r^3 \vec{r} \cdot \vec{\hat{n}} [/itex]

    [tex]
    r^3 \vec{r} \cdot \vec{\hat{n}} = r^3 \vec{r} \cdot \frac{r^3 \vec{r}}{r^3 r} = \frac{r^2 r^3}{r} = r^4
    [/tex]

    so...

    [tex]
    \int \int_S r^3 \vec{r} \cdot \vec{\hat{n}} dS = \int \int_S r^4 dS \\
    = \int_0^{2 \pi} d \phi \int_0^{\pi} R^4 R^2 \sin{\theta} d \theta \\
    = R^6 [\phi]_0^{2 \pi} [\cos{\theta}]_0^{\pi} = 4 \pi R^6
    [/tex]
     
    Last edited: Oct 21, 2015
  2. jcsd
  3. Oct 21, 2015 #2

    Geofleur

    User Avatar
    Science Advisor
    Gold Member

    Well, for starters, I don't think that ## \nabla \cdot r^3 \mathbf{r} = 6r^2 ##.
     
  4. Oct 21, 2015 #3
    That is a given though, it is part of the question. It says, "given that ## \nabla \cdot r^3 \mathbf{r} = 6r^2 ## ..."

    I am assuming that the vector r=(x,y,z) as that is what is used throughout the coursework.
     
  5. Oct 21, 2015 #4

    Geofleur

    User Avatar
    Science Advisor
    Gold Member

    First, note that, if ## r ## has the dimension of length (##L##), then ##r^3 \mathbf{r} ## must have the dimension of ## L^4 ##. But the derivative has the dimension of ## 1 / L ##, so the result must have the dimension of ## L^3 ##, not ## L^2 ##. Second, you can just calculate it. Hint: Write ## r^3 \mathbf{r}## as ##r^3(x\mathbf{i} + y \mathbf{j} + z\mathbf{k}) ## and note that, for example, ## \frac{\partial r}{\partial x} = \frac{x}{r} ##.
     
  6. Oct 21, 2015 #5


    EDIT: Yes I have just calculated it, and i do get [itex]6r^3[/itex]!!
     
    Last edited: Oct 21, 2015
  7. Oct 21, 2015 #6

    Geofleur

    User Avatar
    Science Advisor
    Gold Member

    Even the greatest books have been known to contain errors. Besides, if you do the problem with the correct value of ## \nabla \cdot \mathbf{F} ##, everything works out the way it's supposed to.
     
  8. Oct 21, 2015 #7
    Yes, see my edit above. I get [itex[6r^3[/itex] ! Haha. My lecturer may have told us about the error, but if he did, I was either not there or didnt hear it! haha

    Thanks!!
     
  9. Oct 21, 2015 #8
    Yes I can see now that the LHS would also equal [itex]4 \pi R^6[/itex] when using the correct value for the divergence of F. No wonder they were not equal to eachother! I should have realised the mistake myself to be fair, but this was my first actual problem using Gauss' Theorem, thanks for your help.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Gauss' Theorem - Divergence Theorem for Sphere
  1. Gauss' Theorem (Replies: 4)

  2. Gauss's theorem (Replies: 8)

  3. Gauss's Theorem (Replies: 4)

  4. Gauss's Theorem (Replies: 9)

  5. Gauss' Theorem (Replies: 12)

Loading...