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posted by  princess a on 8/2/2009 6:39:07 PM  |  status: Closed  |  Earned Karma: 48

Divergence Theorem - lifesaver rating

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Use the Divergence Theorem to evaluate the following integral and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
int_S int <b>F</b>*<b>N</b> text( )dS

F(x,y,z) = 2(xi + yj + zk)
S: z = 0, z = sqrt(4-x^2-y^2)
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posted by lawrence_tutor on 8/2/2009 6:51:23 PM  |  status: Live
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The divergence of the vector field is the sum of the partial derivatives, which are all 2, so the divergence is 6.  The divergence theorem says that the flux integral is same as the integral of the divergence over the 3-dimensional region whose boundary is the surface.  Since the divergence is constant, that integral is just 6 times the volume of the region.  As the region is a hemisphere of radius 2, its volume is (2/3)*pi*r^2 = (8/3)*pi.  Multiplying gives 16*pi.
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posted by princess a on 8/2/2009 8:02:20 PM  |  status: Live
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The answer is actually 32π - I followed this model.
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