Divergence Theorem

**Markov** · October 3rd 2011, 10:08 AM

Let $S$ the surface $x^2+y^2-(z-6)^2=0$ for $3\le z\le 6.$ Graph $S$ and graphically indicate an orientation for $S$ and compute the whole flux through $S$ of the field $\bold F(x,y,z)=(x(3-z),y(3-z),(3-z)^2).$

I can apply the divergence theorem but I'm struggled with the bounds, I don't know the triple integral should look like.

How can be done by using a line integral? I'd have to parametrize the surface, but I'd like to know how to set up the integral!

Any help will be appreciated!

**TheEmptySet** · October 3rd 2011, 12:51 PM

Originally Posted by Markov

Let $S$ the surface $x^2+y^2-(z-6)^2=0$ for $3\le z\le 6.$ Graph $S$ and graphically indicate an orientation for $S$ and compute the whole flux through $S$ of the field $\bold F(x,y,z)=(x(3-z),y(3-z),(3-z)^2).$

I can apply the divergence theorem but I'm struggled with the bounds, I don't know the triple integral should look like.

How can be done by using a line integral? I'd have to parametrize the surface, but I'd like to know how to set up the integral!

Any help will be appreciated!

First be careful with the divergence theorem as the cone is not closed and so the divergence theorem does not apply. Note however that the value of the vector field on the flat surface is zero so it wont contribute to the flux and you can close the surface and then apply the divergence theorem.

My guess is that you having problems parameterizing this becuase it is not centered at the origin. We can fix that by shifting everything down 6 units

$z \mapsto z+6$ This gives the equivilent problem

S $x^2+y^2-z^2=0$ for $-3 \le z \le 0$

and the vectorfield is

$\mathbf{F}(x,y,z)=<x(3-(z+6)),y(3-(z+6)),(3-(z+6))^2> \newline =<-x(z+3),-y(z+3),(z+3)^2>$

Now the parametric form of the cone can be found using Cylindrical coordinates or spherical coordinates.

**Markov** · October 3rd 2011, 02:55 PM

Yes, I was thinking the same thing. So I don't have a good parametrization, what could be yours? After that what's the set up for the integral?

By applying the divergence theorem I'm struggling a bit with the bounds, can you give me a hand?

**TheEmptySet** · October 4th 2011, 10:42 AM

Originally Posted by Markov

Yes, I was thinking the same thing. So I don't have a good parametrization, what could be yours? After that what's the set up for the integral?

By applying the divergence theorem I'm struggling a bit with the bounds, can you give me a hand?

The simplist way is to use cylindrial coordinates and you have the equation

$z^2=x^2+y^2$ Just plug the definition of cylindrical coordinates into this equation and it should all work itself out nicely. Give this a shot and see what happens. Warning be very careful with the plus or minus.

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