# divergence theorem

• Jan 30th 2007, 03:46 PM
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.
• Jan 30th 2007, 04:53 PM
ThePerfectHacker
Quote:

Originally Posted by Adebensjp05
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)
$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} 2x+2y+2z dz\, dy\, dx$
• Jan 30th 2007, 05:51 PM
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?
• Jan 30th 2007, 06:05 PM
ThePerfectHacker
Quote:

Originally Posted by Adebensjp05
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).
• Jan 30th 2007, 06:16 PM