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Math Help - surface integrals

  1. #1
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    surface integrals

    Consider the hemispherical volume V defined by:
    x^2 + y^2 + z^2 =< a^2, z>=0, and denote its closed surface by S.
    Define the vector field f=(-y, x, z^2)

    a) Evaluate the net outward flux of f through S by direct integration.

    {to parameterise the curved surface use spherical polar coordinates}

    plz help me its due in a few hours
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  2. #2
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    Quote Originally Posted by martinr View Post
    Consider the hemispherical volume V defined by:
    x^2 + y^2 + z^2 =< a^2, z>=0, and denote its closed surface by S.
    Define the vector field f=(-y, x, z^2)

    a) Evaluate the net outward flux of f through S by direct integration.

    {to parameterise the curved surface use spherical polar coordinates}

    plz help me its due in a few hours
    The vector-function \bold{G}: U\mapsto \mathbb{R}^3 defined by \bold{G}(\theta , \phi) = (a\sin \phi \cos \theta , a\sin \phi \sin \theta , a\cos \theta) where U = [0,2\pi]\times [0,\pi/2) will parametrize the upper hemi-sphere.

    Then, \int_V \bold{F} \cdot d\bold{S} = \pm \int_U (-y,x,z^2) \cdot \left( \partial_{\theta} \bold{G}\times \partial_{\phi} \bold{G} \right) .
    Where \pm depends whether the parametrization was inward or outward. Just pick a vector and check which was it goes.
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