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: Identity proof using Stoke's Theorem

  1. May 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Show using Stoke's Theorem that gif.latex?\bg_white%20\iint_S%20(\nabla%20f)\times%20d\vec%20S%20=%20-\oint_C%20fd%20\vec%20r.gif

    S is an open surface with boundary C (a space curve). [tex]f(\vec r)[/tex] is a scalar field.

    2. Relevant equations

    Stoke's theorem [tex]\iint_S (\nabla\times \vec F) \cdot d\vec S = \int_C F \cdot d\vec r[/tex]

    3. The attempt at a solution

    Thus far I've tried evaluating [tex] (\nabla f)\times d\vec S[/tex] by taking components in various ways and then trying to force it to become something that I can take the curl of... with no success at all. I'm completely stumped now.
  2. jcsd
  3. May 16, 2010 #2


    User Avatar
    Homework Helper
    Gold Member

    Try applying Stokes' theorem to the vector function [itex]\textbf{F}\equiv f\textbf{c}[/itex], where [itex]\textbf{c}[/itex] is any constant vector.:wink:
  4. May 16, 2010 #3
    Well, what I get here is

    [tex] \iint_S \vec c\cdot\nabla f \times d\vec S = -\oint \vec c \cdot fd\vec r[/tex]

    And I'm not entirely sure whether I can just "cancel" the c dot product...

    (should be a vector c there, by the way)
    Last edited: May 16, 2010
  5. May 16, 2010 #4


    User Avatar
    Homework Helper
    Gold Member

    Well, since [itex]\textbf{c}[/itex] is a constant vector, you can certainly pull it out of the integrals and say

    [tex]\vec c\cdot\left(\iint_S \nabla f \times d\vec S\right) = \vec c\cdot\left(-\oint fd\vec r\right)[/tex]

    [tex]\implies \vec c\cdot \left(\iint_S \nabla f \times d\vec S+\oint fd\vec r\right)=0[/tex]

    And since this is true for arbitrary, constant [itex]\vec{c}[/itex], what must you conclude?
  6. May 16, 2010 #5
    Yeah, those last steps were fine for me, I just wasn't sure if that even with c a constant vector that it was "allowed" to just pull it out like so. Thanks for the help.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook