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A challenging vector field path integral

  1. Dec 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Evaluate ∫F dot ds

    2. Relevant equations

    F = < 1 - y/ (x^2 + y^2) , 1 + x/(x^2 + y^2) , e^z >

    C is the curve z = x^2 + y^2 -4 and x + y + z = 100

    3. The attempt at a solution

    I don't think Stokes theorem applies since the vector field is undefined at the origin, so I'm trying a path integral according to ∫F(c(t) dot c'(t) dt for path c. The problem is that I combined the curve equations into a completed square that gave me a parameterization that I don't see how to integrate.

    x^2 + y^2 - 4 = 100 - x - y
    (x+1/2)^2 + (y+1/2) ^2 = 104.5

    x = √104.5 cos t - 1/2
    y = √104.5 sin t - 1/2
    z = 100 - √104.5 cos t - √104.5 sin t

    for t from 0 to 2∏

    The resulting integral is an unholy mess. Am I missing something?

    Thank you all for a great forum!
     
  2. jcsd
  3. Dec 7, 2013 #2

    vela

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    As long as the surface you're integrating over doesn't contain the origin, you can use Stoke's theorem.

    Edit: Oh, wait, I see you actually need to worry about the entire z-axis.
     
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