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Using Stoke's Theorem to determine plane area enclosed by simple closed plane curve

  1. Nov 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Let C be a simple closed plane curve in space. Let n = ai+bj+ck be a unit vector normal to the plane of C and let the direction on C match that of n. Prove that

    (1/2)∫[(bz-cy)dx+(cx-az)dy+(ay-bx)dz]

    equals the plane area enclosed by C.

    What does the integral reduce to when C is in the xy-plane?


    2. Relevant equations

    Stoke's Theorem

    F.ds=∫(∇×F).dS


    3. The attempt at a solution

    I really have no idea where to start. Any help would be much appreciated.
     
  2. jcsd
  3. Nov 30, 2012 #2

    LCKurtz

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    Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

    Why don't you start by calculating ##\nabla \times \vec F## and plug it in the right side of Stokes' Theorem?
     
  4. Nov 30, 2012 #3
    Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

    I don't know what F is
     
  5. Nov 30, 2012 #4

    LCKurtz

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    Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

    You are given a line integral, written in the form ##\oint \vec F\cdot d\vec R##. Can't you pick ##\vec F## out of that?
     
  6. Nov 30, 2012 #5
    Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

    I think I might have been reading the question incorrectly (well hopefully). I feel like an idiot. I'll give it a go and hopefully it works out. Thanks LCKurtz
     
  7. Nov 30, 2012 #6
    Re: Using Stoke's Theorem to determine plane area enclosed by simple closed plane cur

    Got it out

    Thanks again for your help LCKurtz
     
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