Divergence theorem over a hemisphere

In summary, the conversation discusses a problem that can be solved using the divergence theorem by adding a unit disc on the bottom and then subtracting it again. The total integral from the divergence theorem is 4pi/3, and when calculating the surface integral over the upper hemisphere, the answer should be 7pi/3 instead of 7pi/6.
  • #1
jonwell
11
0
I was told this problem could be done with divergence theorem, instead of as a surface integral, by adding the unit disc on the bottom, doing the calculation, then subtracting it again.

Homework Statement



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Homework Equations



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The Attempt at a Solution



for del . f I get i + j = 2. Which makes the integral equal twice the volume of the hemisphere, or 4/3 pi. Now I'm supposed to subtract the unit disc, but I get pi when I calculate that surface, which leaves me with 1/3 pi. The answer should be 7/6 pi.

Thanks :)
 
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  • #2
The outward pointing normal to the disk is -k. So F.(-k) is -1 and the contribution from the disk is -pi. The total integral from the divergence theorem is 4pi/3 (as you said). I think that makes the integral over the upper hemisphere 7pi/3, doesn't it? Not 7pi/6?
 
  • #3
Ya, I got that a couple times too (after I remembered the orientation), however the answer I was provided is 7/6 pi. That could be wrong. I'll re-do the surface integral the other way and see what I come up with I guess.
 
  • #4
Good idea! Let me know what you get.
 
  • #5
well, I'm still getting 7pi/3, so I'm going to assume that the two of us combined are smarter than the given answer ;) Thanks for your help!
 

FAQ: Divergence theorem over a hemisphere

What is the Divergence theorem over a hemisphere?

The Divergence theorem over a hemisphere is a mathematical concept that relates the flux (flow) of a vector field through a closed surface over a hemisphere to the volume integral of the divergence of that vector field over the hemisphere.

Why is the Divergence theorem over a hemisphere important?

The Divergence theorem over a hemisphere is important because it allows for the calculation of the flux through a closed surface over a hemisphere by only considering the divergence of the vector field, rather than having to calculate the flux directly. This simplifies many calculations in physics and engineering.

How is the Divergence theorem over a hemisphere derived?

The Divergence theorem over a hemisphere is derived from the more general Divergence theorem, which relates the flux through a closed surface to the volume integral of the divergence over the volume enclosed by that surface. By considering a hemisphere as a special case of a closed surface, the Divergence theorem over a hemisphere can be derived.

What are some real-world applications of the Divergence theorem over a hemisphere?

The Divergence theorem over a hemisphere has many applications in physics and engineering, such as calculating the flow of fluids through a hemisphere-shaped container or the electric flux through a hemisphere-shaped capacitor. It is also used in weather forecasting to model the flow of air over a hemisphere-shaped region.

Are there any limitations to the Divergence theorem over a hemisphere?

While the Divergence theorem over a hemisphere is a powerful tool for calculating flux, it is limited to cases where the surface of interest is a hemisphere. It also assumes that the vector field is continuous and differentiable over the surface. In some cases, other mathematical theorems may need to be used in conjunction with the Divergence theorem over a hemisphere to fully understand the flux through a given surface.

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