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Math Help - Divergence Theorem and Flux

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    Senior Member AllanCuz's Avatar
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    Divergence Theorem and Flux

    Note: I've attached images of my work at the bottem of this post.

    I've calculated the flux through a given surface by using The
    Divergence Theorem and by using the regular flux method. These
    methods give different results, however.

    This leads me to assume one of the following is true:

    Note: N is the unit vector outter normal to S (probably didn't have
    to be said, but I didnt include it in my question on the paper!)

    1) My math is wrong in one or both of the cases
    2) One of these methods doesn't apply here
    3) I didn't fully complete the regular flux method and I didnt
    account for the s4 side (as depicted in the picture in the second
    image) but I should have.

    The equation used is: NdS = +/- (-dF/dx - dF/dy + K)
    Of course for the sides that are along the cordinate axis
    it is easiar to compute F*N because N in that case would
    be along one of the cordinate axes and simply be a unit vector
    along the axis.

    I believe that the surface is entirely smooth (1-piece)
    and can be evaluated using one instance of that equation.
    Also I thought the question is asking for the Flux through
    (i.e. only the flux out of the top surface) This is my
    original assumption. However...

    I'm fairly sure that my original assumption was wrong and that I
    need to compute all the various sides (diagram shown in the second
    link). Of course, most of these sides will result in 0 flux due to
    the dot product of perpendicular vectors. But this is computed
    through the divergence theorem easily enough. So I think that the
    way I did it with the divergence theorem is the correct way. Yes?

    Calculated Flux through just the upper surface

    http://img163.imageshack.us/img163/6020/question0011.th.jpg

    Calculated via the Divergence Theorem

    http://img28.imageshack.us/img28/1120/question10review0001.th.jpg


    Cheers
    Last edited by AllanCuz; April 9th 2010 at 01:22 PM. Reason: Changed second image (had a mistake!)
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