1. Not finding help here? Sign up for a free 30min 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!

Multivariable Calc help

  1. Jul 12, 2005 #1

    cronxeh

    User Avatar
    Gold Member

    Hey can you guys check my answer.

    Question: Use the Divergence Theorem to calculate the Flux of the vector field F(x,y,z)=xi + y^2j - zk through the unit sphere centered at the origin with the outward orientation

    Solution: div(F) = 1 + 2y - 1 = 2y
    Flux = [tex]\int_{W} div(F) dV = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 2y \ dxdydz = \int_{0}^{1} \int_{0}^{1} 2y \ dydz = \int_{0}^{1} 1 \ dz = 1 [/tex]

    So for another method would I have to use [tex]r=x^2 + y^2 + z^2[/tex] obtain the [tex]dr=2x + 2y + 2z[/tex] and do [tex] \int_{R} F dr = \int_{S} (xi + y^2j -zk)(2x+2y+2z)dA=\int_{S} (2x^2 + 2y^3 - 2z^2)dA = \int_{0}^{1} \int_{0}^{1} 2/3 + 2y^3 - 2z^2 \ dydz = \int_{0}^{1} 2/3 + 1/2 - 2z^2 dz = 2/3 + 1/2 - 2/3 = \frac{1}{2} [/tex]

    Is this method incorrect or was the first one incorrect? Are they both wrong? :confused:
     
  2. jcsd
  3. Jul 12, 2005 #2
    ermm... wouldn't it be easiest to use spherical coordinates?


    edit: and in that second method, why are you calculating a line integral?

    gauss' theorem doesn't have line integrals in there at all.

    look up what gauss' theorem tell you...

    i imagine you wanted to solve the problem the easy (integrating the divergence) way and the harder (what integrating the divergence is equal to :wink: ) way.


    so to answer your question... both methods are incorrect! :eek:
     
    Last edited: Jul 12, 2005
  4. Jul 12, 2005 #3
    yeah, the way you have written the limits of integration, you are taking the divergence of that vector field through the unit cube in your first method. (haven't bothered with your second method--get your first one fixed!)
     
  5. Jul 12, 2005 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Do you have any idea what regions you're supposed to be integrating over, for either calculation?
     
  6. Jul 12, 2005 #5
    and is it just me or is he trying to use stokes' theorem in that second attempt?

    :eek:

    edit: no...closer look tells me that... it's not even that.

    your second method is way off base.


    look back at your textbook and see the two ways of calculating flux.
     
    Last edited: Jul 12, 2005
  7. Jul 12, 2005 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The second method is surely supposed to be the definition of flux: [itex]\iint_{\delta R} \vec{F} \cdot \hat{n} \, dA[/itex].
     
  8. Jul 12, 2005 #7
    yeah, that's the other--probably harder--way to do the problem. except... it looks like he was trying to use a line integral or...something. :confused:
     
  9. Jul 12, 2005 #8

    cronxeh

    User Avatar
    Gold Member

    Ok I just realized my mistake for divergence method ( I hope )

    div F = 2y

    Should the flux be: [tex]\int_{W} div(F) \ dV =[/tex]

    [tex]\int_{-1}^{1} \int_{-sqrt(1-x^2)}^{sqrt(1-x^2)} \int_{-sqrt(1-x^2-y^2)}^{sqrt(1-x^2-y^2)} 2y \ dzdydx[/tex]
     
  10. Jul 12, 2005 #9

    cronxeh

    User Avatar
    Gold Member

    And in spherical coordinates:

    Flux = [tex]\int_{0}^{2pi} \int_{0}^{pi} \int_{0}^{1} 2y \ p^2 \ sin(phi) \ dp \ dphi \ dtheta[/tex]
     
  11. Jul 12, 2005 #10

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Yah, but don't forget you should convert y to spherical coordinates too.
     
  12. Jul 12, 2005 #11

    cronxeh

    User Avatar
    Gold Member

    that would be 2y ->> 2(p)sin(theta)sin(phi) ?
     
  13. Jul 12, 2005 #12

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Because in spherical coordinates

    x= ρcosθsinφ
    y= ρsinθsinφ
    z= ρcosφ
     
  14. Jul 12, 2005 #13
    yeah, good going.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Multivariable Calc help
  1. Calc Help (Replies: 1)

Loading...