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Math Help - divergence

  1. #1
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    divergence

    Hi guys, any help with this would be greatly appreciated!


    Illustrate (verify) the Divergence Theorem with the force function
    F (x,y,z) = <1, 2, z^3> where the surface S is the cylinder x^2 + y^2 = 16, 0z6.

    left side of Gauss's Th =

    right side of Gauss's Th =
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  2. #2
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    Quote Originally Posted by boousaf View Post
    Hi guys, any help with this would be greatly appreciated!


    Illustrate (verify) the Divergence Theorem with the force function
    F (x,y,z) = <1, 2, z^3> where the surface S is the cylinder x^2 + y^2 = 16, 0z6.

    left side of Gauss's Th =

    right side of Gauss's Th =
    \int \int_{R_{xy}} \int_{z=0}^{z=6} 3z^2 \, dz \, dy \, dx = (6^3) (\pi 4^2)

    since  \nabla \cdot F = 3z^2 and R_{xy} is a circle of radius 4.


    By symmetry the only non-zero contribution of the flux of F through S is through the top of the cylinder ...... (You can check this - using cylindrical coordinates for the flux through the curved part of the cylinder is probably easiest but it can be done using cartesian coordinates too). Note that [tex]\vec{dS} = k dx dy on the ends of the cylinder.

    Then the flux integral becomes \int \int_{x^2 + y^2 = 16} z^3 \, dx \, dy = \int \int_{x^2 + y^2 = 16} 6^3 \, dx \, dy = (6^3) (\pi 4^2).
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  3. #3
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    thanks again mr. fantastic!!
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