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: Stokes Theorem

  1. May 10, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that

    ## \oint_{\partial S} ||\vec{F}||^2 d\vec{F} = -\int\int_S 2 \vec{F}\times d\vec{A} ##

    2. Relevant equations


    ##\nabla \times (||\vec{F}||^2 \vec{k}) = 2\vec{F} \times \vec{k} ##

    For ##\vec{k} ## constant i.e. ## \nabla \times \vec{k} = 0 ##

    Stokes Theorem

    ##\oint_{\partial S} \vec{B} \cdot d\vec{x} = \int\int_S (\nabla \times \vec{B})\cdot d \vec{A} ##

    3. The attempt at a solution

    So I need to use that identity ##\nabla \times (||\vec{F}||^2 \vec{k}) = 2\vec{F} \times \vec{k} ##

    The problem is that Stokes theorem is in a different form. The constant vector here I think is the k=dA.

    I really can't think of what to do
    Last edited: May 10, 2012
  2. jcsd
  3. May 10, 2012 #2


    User Avatar
    Gold Member

    Well, [itex]\vec F[/itex] is the vector field, but i'm not sure what [itex]||\vec F||[/itex] represents in the original equation.
  4. May 10, 2012 #3
    I made an error. It is a squared term.
  5. May 10, 2012 #4


    User Avatar
    Gold Member

    Assuming that there are no more mistakes in your first post, [itex]\vec k[/itex] represents the outward unit normal vector from the surface, S.
  6. May 10, 2012 #5
    Maybe you should use Stokes' theorem for each component of the original vector-valued integral
    [tex] \vec{I}= \oint ||\vec{F}||^2 d\vec{F} [/tex]
    [tex] I_x = \oint (||\vec{F}||^2 \vec{e}_x) \cdot d\vec{F} [/tex]
    etc. Now it's in the correct form.
  7. May 10, 2012 #6
    I dont understand this?
  8. May 10, 2012 #7
    [itex]\vec{e}_x [/itex] is the unit vector in x-direction. The lower integral is just the x-component of your full integral. You can calculate this by taking the dot product with [itex] \vec{e}_x [/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook