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:

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,
$\displaystyle \oint _S \oint \bold{F}\cdot \bold{n} dS=\int \int_R \int \mbox{ div }\bold{F} dV$
Thus, all we need to do if find,
$\displaystyle \int \int_R \int \mbox{ div }\bold{F} dV$
Now,
$\displaystyle \mbox{div }\bold{F}=2x+2y+2z=2x+2y+2z$
Thus, by Fubini's theorem, (details omitted)
$\displaystyle \int_0^1 \int_0^{1-x} \int_0^{1-x-y} 2x+2y+2z dz\, dy\, dx$
• 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: