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Math Help - Stoke's Theorem

  1. #1
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    Stoke's Theorem

    In the interest of disclosure, this is a homework problem for which I will
    receive a grade.

    First the problem,

    Let S be a closed surface given by x^4+y^4+z^4=1.

    Evaluate
    <br />
\iint\limits_S {(\nabla  \times F)} \bullet nd\sigma <br />

    Where F=<2xy^2cosz,x+yze^z,xyz> and n is the outer normal.

    Now my dilemma:
    Using Stokes Theorem I believe I can evaluate as a line integral C or as a surface integral S.
    My problem is with the parameterization.
    The surface(x^4+y^4+z^4=1) looks similar to a cube but with rounded corners.
    If I evaluate as a line integral then I am unsure of what parameters to use since this ( at least to me) is not a line I am used to defining. Unlike lets say, a circle, which can be parameterized as x=rcos, y=rsin.

    Alternatively, if i try to evaluate as a surface integral then I think that to parameterize the position vector it would look something like
    r=<x,y,(1-x^4-y^4)^1/4>.
    This method begins to get complicated when I proceed to the next step and try to evaluate the cross product of dr/dx x dr/dy.
    So, finally, I guess I question is-do I evaluate as a line or surface?
    Also, how do parameterize this problem?
    Any help is greatly appreciated.

    Kid
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  2. #2
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by kid funky fried View Post
    In the interest of disclosure, this is a homework problem for which I will
    receive a grade.

    First the problem,

    Let S be a closed surface given by x^4+y^4+z^4=1.

    Evaluate
    <br />
\iint\limits_S {(\nabla \times F)} \bullet nd\sigma <br />

    Where F=<2xy^2cosz,x+yze^z,xyz> and n is the outer normal.

    Now my dilemma:
    Using Stokes Theorem I believe I can evaluate as a line integral C or as a surface integral S.
    My problem is with the parameterization.
    The surface(x^4+y^4+z^4=1) looks similar to a cube but with rounded corners.
    If I evaluate as a line integral then I am unsure of what parameters to use since this ( at least to me) is not a line I am used to defining. Unlike lets say, a circle, which can be parameterized as x=rcos, y=rsin.

    Alternatively, if i try to evaluate as a surface integral then I think that to parameterize the position vector it would look something like
    r=<x,y,(1-x^4-y^4)^1/4>.
    This method begins to get complicated when I proceed to the next step and try to evaluate the cross product of dr/dx x dr/dy.
    So, finally, I guess I question is-do I evaluate as a line or surface?
    Also, how do parameterize this problem?
    Any help is greatly appreciated.

    Kid
    I'm fairly certain this would be best evaluated in its present form. In other words, I wouldn't do the parametric route.

    We are asked for,

     \iint_S curl F \cdot \hat N dS

    We area given off the hop

    F = (2xy^2cosz) \hat i + (x+yze^z) \hat j + (xyz) \hat k

    So we can easily find Curl F in this form. Now what about N and dS? Well...

     \hat N dS = +/- ( - \frac{ dg}{ dx} \hat i - \frac{ dg}{ dy} \hat j + \hat k )

    But I believe we can simpliy this to (after applying strokes twice)

     \hat N dS = \hat k dA

    We then get,

     \iint_S curl F \cdot \hat N dS = \iint_D curl F \cdot \hat k dA

    This should simplify to something simple (remember to evaluate curl F at z=0 ).

    The domain might be a problem but I believe we can do this in cartesian just fine.
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