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Problem with a simple Stoke's Theorem question

  1. May 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Using Stoke's Theorem, evaluate the contour integral:
    [tex]\oint F.dr[/tex]
    as an integral over an appropriately chosen 2 dimensional surface.

    Use F = [tex](e^{x}y+cos\siny,e^{x}+sinx\cosy,ycosz)[/tex] and take the contour C to be the boundary of the rectangle with the vertices (0,0,0), (A,0,0), (A,B,0), (0,B,0) oriented anticlockwise.

    Then evaluate the same integral directly as a contour integral.

    2. Relevant equations
    Stoke's Theorem,

    [tex]\int\int_{S}(curl F).n\;d^{2}A=\oint F.dr[/tex]
    where n is unit normal vector.

    3. The attempt at a solution

    I got Curl F to be (cos z)i by the standard method using a matrix. I set n as -k as the surface is in the xy plane so the normal vector is along the z direction and I got the negative by the right hand rule.

    This gives:


    (-k).(cos z)i is 0

    which makes the double integral nothing

    However when I solve the same thing directly as a contour integral I get an answer of -B.

    Going anti clockwise from the origin, (so from (0,0,0) to (A,0,0) is I, (A,0,0) to (A,B,0) is II, (A,B,0) to (0,B,0) is III and (0,B,0) to (0,0,0) is IV) I get:
    II=[tex]e^{A}B + sinA\;sinB[/tex]

    Which, when added together, gives -B.... I think there must be a glaring error somewhere. I can write up my calculations for the contour integral if that would help solve it... Any help with this would be much appreciated.
  2. jcsd
  3. May 10, 2009 #2


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    Homework Helper
    Gold Member

    Technically, the normal to the surface should be +k as given by the right-hand rule when your path is anticlockwise. Luckily, it didn't affect your answer in this case as -1*0 is still zero..

    Your error is in integral III; recheck that calculation and post your work for it if you can't find the error.
  4. May 11, 2009 #3
    I got it! Thanks a lot. Ha, I somehow messed up with the right hand rule, which is pretty basic, but that wasn't the problem. I just made a copying error and forgot about one of the [tex]e^{0}B[/tex] which should have become B, not nothing. Thanks a lot!
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