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Green's Theorem and zero

  1. Aug 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Use Green's Theorem to evaluate the line integralalone the given positvely oriented curve.

    ∫[tex]_{c}[/tex] sin(y)dx+xcos(y)dy, C is the ellipse x2+xy+y2=4



    2. Relevant equations



    3. The attempt at a solution
    ∫∫(cos(y)-cos(y))dA=∫∫0dA

    Because this ends up being the double integral of zero, does this just mean my answer is zero?
     
  2. jcsd
  3. Aug 3, 2010 #2

    Dick

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    That's right. The line integral around any closed curve will be 0.
     
  4. Aug 3, 2010 #3

    vela

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    Yup.
     
  5. Aug 3, 2010 #4
    Awesome. Thank you both.
     
  6. Aug 3, 2010 #5
    Correction: The line integral around any closed curve will be 0 ONLY if the integral is on a conservative field. otherwise it won't be zero for all cases.
     
  7. Aug 3, 2010 #6

    Dick

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    Mmm. Well, sure. F=(sin(y),x*cos(y)) is conservative. So the integral of F.dr is zero around any closed curve. I didn't mean ANY F. Did that really need a 'correction'?
     
  8. Aug 4, 2010 #7
    It's a subtle point, I wanted EV33 to know that :)
    It is very easy to make that mistake and think that any close line integral is zero.
     
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