Verifying difvergence theorem

1. Jun 4, 2010

zak8000

1. The problem statement, all variables and given/known data
verify that the divergence theorem in 3-d is true for the vector field F(r)=<3x,xy,2xz>
on the cube bounded by the planes x=0 x=1 y=0 y=1 z=0 z=1

2. Relevant equations

3. The attempt at a solution

so fristly div(F)=d/dx(3x)+d/dy(xy)+d/dz(2xz)=3+3x
$$\int$$$$\int$$$$\int$$ 3+3xdxdydz=4.5

now i need to evaluate flux through each faces of the cube seperately so i was just wondering if i am doing this write say i would want to evaluate the top surface of the cube
then i would have to parametrize it so would the following be corret
r(x,y,z)=(3x,xy,1)
dr/dx=(3,y,0)
dr/dy=(0,x,0)
(dr/dx) X (dr/dy) = (0,0,3x)
r(x,y,z).((dr/dx) X(dr/dy))= (3x,xy,1).(0,0,3x) = 3x
$$\int$$$$\int$$ 3x dydx
=3/2

and i have to do the same for all other five surfaces so is this the correct way?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 5, 2010

vela

Staff Emeritus
You're not using the correct r when calculating the normal. You want to describe the surface, not the vector field. For the top face, it would be r=(x, y, 1).

3. Jun 5, 2010

ok thanks