Divergence Theorem Help - Flux of Vector A Through V

[Nima]

Consider the volume V bounded below by the x-y plane and above by the upper half-sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 1

Given vector field: A = xi + yj + zk

Use the divergence theorem to calculate the flux of A out of V through the spherical cap on the cylinder.

Really stuck!

Could someone please give me tips on how to answer these sort of questions? I get really stuck, like firstly replacing dS in the surface integral, and what equations to use, etc. I couldn't even get going with this one, my only idea was just calculating the flux through the upper half sphere ... how does the cylinder equation come into it?

Thanks

 

[0rthodontist]

Can you visualize what the region is? It is like a grain silo. The sphere has radius 2 and the cylinder only has radius 1.

I think they may want you to approach it by calculating three things:
1. the flux out of the base of the silo
2. the flux out of the sides of the silo
3. the total divergence within the silo
You can then use the divergence theorem to find the flux out of the cap of the silo. I do not really know if this is simpler than calculating the flux directly. You can use cylindrical coordinates which will simplify it.

 

[pocoman]

Too complex if you calculate the flux directly.
The divergence theorem states that:
[tex] flux = \iint_S AdS = \iiint_V \nabla A dV [/tex]
Given vector field A=xi+yj+zk =r and easily find the form of
[tex] \nabla A [/tex]
Applying the divergence theorem, you will quickly find the solution.

 

[FrogPad]

It would probably make your life easier to compute the volume in spherical coordinates.

And the divergence theorem is:

[tex] flux = \phi = \iint_S \vec F \cdot d\vec S = \iiint_{\partial V} \nabla \cdot \vec F \,\,dV [/tex]

Remember that [itex] d\vec S = \hat n dS [/itex] and that [itex] \partial V [/itex] is the bounds on the integral that contain your "object" that you are finding the flux through.

It's important to remember that the dot product returns a SCALAR. You are integrating a scalar function NOT a vector. So when you do:

[tex] \nabla \cdot \vec A [/tex]. You get a scalar function which happens to just turn out to be a constant right? So really you are trying to find something of the form:

[tex] 3 \iiint dV [/tex] which is really just a constant times a volume. You could probably just find the volume geometrically without any calculus. Try setting up the problem step by step.

 

[HallsofIvy]

Are you really required to use the divergence theorem? This looks like exactly the kind of problem Stoke's theorem is designed for.

 

[siddharth]

HallsofIvy, I don't understand how you could apply Stoke's theorem in this case to calculate the net flux.