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!

Another Stokes' Theorem Problem

  1. Dec 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Verify Stokes' theorem for the given surface S and boundary dS and vector fields F

    S = x2+y2+z2, z≥0
    dS= x2+y2=1

    F = <y,z,x>

    2. Relevant equations

    Stokes' theorem:
    ∫∫(∇×F)dS = ∫F⋅ds

    3. The attempt at a solution
    1. Curl of F:

    ∇×F = <-1,-1,-1>

    2. After getting the curl, I just treated this as a surface integral, ∫∫F⋅dS = ∫∫F⋅(Tu×Tv)dudv

    I parametrized the sphere thusly,

    x = sinφcosθ
    y=sinφsinθ
    z=cosφ

    Tφ= <cosφcosθ, cosφsinθ, -sinφ>
    Tθ= <-sinφsinθ, sinφcosθ, 0>

    Tφ×Tθ = <sin2φcosθ, sin2φsinθ, sinφcosθ>

    Am I doing this right so far? I asked a friend what he would do, and this is what he had for dS:
    <dydz, dxdz, dxdy> (Idk how to get this).

    So then he got:

    ∫∫(∇×F)⋅dS = ∫∫(<-1, -1, -1>⋅<dydz, dxdz, dxdy>) = -∫∫dydz+dxdz+dxdy, but I don't know how to integrate that...
     
  2. jcsd
  3. Dec 17, 2014 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I would ignore your friend's comments. Just continue what you are doing. You are now going to calculate$$
    \iint \langle -1,-1,-1\rangle \cdot T_\phi \times T_\theta~d\phi d\theta$$with appropriate limits. Be sure to check whether you need a minus sign for orientation or not.
     
  4. Dec 17, 2014 #3

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    There's a common factor of ##\sin\phi##. If you pull it out front, you have ##\sin\phi \langle \sin\phi \cos\theta, \sin\phi \sin\theta, \cos\phi \rangle##. If you recognize that vector, you should be able to convince yourself you're on the right track.
     
    Last edited: Dec 19, 2014
  5. Dec 23, 2014 #4

    rude man

    User Avatar
    Homework Helper
    Gold Member

    This defines a surface?
    A half-sphere of radius sqrt(S)?
     
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: Another Stokes' Theorem Problem
  1. Stokes' Theorem problem (Replies: 27)

  2. Stokes theorem problem (Replies: 1)

Loading...