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!

Surface integrals

  1. Feb 9, 2007 #1
    Please Help! Surface integrals

    I am wondering if someone can help me with the following? I am asked to evaluate ∫∫F∙dS where F(x,y,z) = z^2xi + (1/3y^3 +tanz)j + (x^2z+y^2)k and S is the top half of the sphere x^2+y^2+z^2 = 1.

    ∫∫F∙dS = ∫∫∫divFdV. Here, div F = x^2+y^2+z^2. I know that S is not a closed surface and so you would need to evaluate S as the difference between 2 surfaces, S1 as the closed surface that is the top half of the sphere and S2 as the disk that is x^2+y^2≤1 where the orientation is downwards. So, evaluating div F over the whole top half of the sphere I got 2pi/5.

    But I am wondering how I would evaluate ∫∫∫x^2+y^2+z^2 over the disk x^2+y^2≤1? Could I convert to spherical coordinates and I could simplify the expression of the integrant to ρ^4sinφ, but I am wondering what φ would be in this instance? Thanks so much!
  2. jcsd
  3. Feb 9, 2007 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    You wouldn't integrate a volume integral over a surface!

    [itex]\int\int\int (x^2+ y^2+ z^2)dzdydx[/itex] is integrated over the volume- the half-ball- not the surface. If you are using cartesian coordinates, integrate with x from -1 to 1, y from [itex]-\sqrt{1- x^2}[/itex] to [itex]\sqrt{1- x^2}[/itex], z from [itex]-\sqrt{1- x^2- y^2}[/itex] to [itex]\sqrt{1- x^2- y^2}[/itex]. It's simpler in cylindrical coordinates: integrate with r from 0 to 1, [itex]\theta[/itex] from 0 to [itex]\2 pi[/itex], z from 0 to [itex]\sqrt{1- r^2}[/itex]. And, of course, it's much simpler in spherical coordinates: integrate with r from 0 to 1, [itex]\theta[/itex] from 0 to [itex]2\pi[/itex], [itex]\phi[/itex] from 0 to [itex]\pi/2[/itex].
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Surface integrals
  1. Surface integral (Replies: 6)

  2. Surface integrals (Replies: 10)