Divergence Theorm example for 28 Nov 12:00

[debian]

Homework Statement

Let S be a smooth surface enclosing the volume V, and let $\vec{n}$ to be the unit outward normal. Using the Divergence theorem show that:


∫∫ x $\vec{r}$ ° $\vec{n}$ dS = 4 * ∫∫∫ x dV,

where $\vec{r}$=(x,y,z)

Homework Equations

Divergence theorem

http://www.math.oregonstate.edu/home...rg/diverg.html

The Attempt at a Solution

I tried to change the form of the those two equations to the form stated in divergence theorem and then to compare the u (or F as in link above), but the u (F) on the left hand side is never equal to this on the right.

 

[clamtrox]

Your link is broken. But let's suppose the divergence theorem says $$\iint (\vec{F} \cdot \vec{n}) dS = \iiint \nabla \cdot \vec{F} dV.$$ Now you are given $\vec{F}= x \vec{r}$. Can you calculate $\nabla \cdot \vec{F}$?

 

[debian]

$\nabla$ $\cdot$ $\vec{F}$ = (d/dx, d/dy, d/dz) $\cdot$ (x^2,xy,xz) = 2x+x+x=4x

 

[clamtrox]

$\nabla$ $\cdot$ $\vec{F}$ = (d/dx, d/dy, d/dz) $\cdot$ (x^2,xy,xz) = 2x+x+x=4x

Good job. :)