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Math Help - divergence theorem

  1. #1
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    divergence theorem

    Can someone just help me set up the following integrals?

    For the region R in the first octant bounded above by the plane x+y+z=1 and the surface S which is the boundary of this region, calculate for the vector valued function F=(x^2+y^2)i+(y^2+z^2)j+(z^2+x^2)k:

    (i) triple integral (div F) dV
    (ii) double integral (F*n) dS where n is the outward pointing unit normal.

    Thank you for any help.
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  2. #2
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    Quote Originally Posted by Adebensjp05 View Post
    Can someone just help me set up the following integrals?

    For the region R in the first octant bounded above by the plane x+y+z=1 and the surface S which is the boundary of this region, calculate for the vector valued function F=(x^2+y^2)i+(y^2+z^2)j+(z^2+x^2)k:

    (i) triple integral (div F) dV
    (ii) double integral (F*n) dS where n is the outward pointing unit normal.
    First I like to mention that by Gauss' theorem,
    \oint _S \oint  \bold{F}\cdot \bold{n} dS=\int \int_R \int \mbox{ div }\bold{F} dV
    Thus, all we need to do if find,
    \int \int_R \int \mbox{ div }\bold{F} dV
    Now,
    \mbox{div }\bold{F}=2x+2y+2z=2x+2y+2z
    Thus, by Fubini's theorem, (details omitted)
     \int_0^1 \int_0^{1-x} \int_0^{1-x-y} 2x+2y+2z dz\, dy\, dx
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  3. #3
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    Yeah, so my assignment paper has (i) and (ii) as two different questions and answers. Is there a difference?

    By Gauss' Theorem (divergence theorem), they appear to be the same?
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  4. #4
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    Quote Originally Posted by Adebensjp05 View Post
    Yeah, so my assignment paper has (i) and (ii) as two different questions and answers. Is there a difference?

    By Gauss' Theorem (divergence theorem), they appear to be the same?
    They are the same. And you can compute the double line integral through Gauss' theorem. Unless, the problem specifices to do it through parametrization (the standard way to compute line integrals).
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  5. #5
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    Are they only the same because we are talking about a plane?

    Some problems in my text appear different when the figure is a sphere or a cylinder.
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