How to use the divergence theorem to solve this question

In summary, the correct answer for the problem is ##\frac{\pi a^2 h} 2## by using the standard approach. However, when using the divergence theorem, a different answer is obtained due to not considering the condition that the surface is only to the right of the xz plane. The volume calculated is only half of the cylinder, and additional surfaces are needed to close the surface where the integral contribution is zero.
  • #1
Leo Liu
353
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Homework Statement
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Relevant Equations
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The correct answer is ##\frac{\pi a^2 h} 2## by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows:
$$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$
$$= \iiint_D \frac{\partial y}{\partial y}\, dV$$
$$=\iiint_D \, dV=V_{cylinder}=\pi a^2h$$
Where did I go wrong?
 
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  • #2
Is the surface integral over a closed surface?
 
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Likes Leo Liu
  • #3
Your volume is only half the cylinder.
(After you add the surfaces needed to close the surface where the integral contribution is zero.)
 
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Likes PeroK and Leo Liu
  • #4
PeroK said:
Is the surface integral over a closed surface?
Nope. But IMO the flux passing through the led of the cylinder and the bottom of the cylinder is 0.
 
  • #5
Orodruin said:
Your volume is only half the cylinder.
(After you add the surfaces needed to close the surface where the integral contribution is zero.)
I am sorry. I don't get it.
 
  • #6
Leo Liu said:
I am sorry. I don't get it.
You've calculated the volume of the whole cylinder. The surface is only half the cylinder.
 
  • #7
PeroK said:
You've calculated the volume of the whole cylinder. The surface is only half the cylinder.
1626972009451.png

Okay I see -- I didn't see the condition that the surface is to the right of the xz plane. Thank you! :smile:
 

FAQ: How to use the divergence theorem to solve this question

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical theorem that relates the flux of a vector field through a closed surface to the divergence of the field within the volume enclosed by the surface. It is a powerful tool in solving problems involving vector fields in three-dimensional space.

2. How do I use the divergence theorem to solve a question?

To use the divergence theorem, you first need to identify the vector field and the closed surface in the question. Then, calculate the divergence of the vector field and the flux of the field through the surface. Finally, apply the divergence theorem equation, which states that the flux through the surface is equal to the triple integral of the divergence of the field within the volume enclosed by the surface.

3. What types of problems can the divergence theorem be used to solve?

The divergence theorem can be used to solve a wide range of problems involving vector fields, such as calculating the flow of fluid through a closed surface, determining the electric flux through a closed surface, and finding the net force on a solid object due to a surrounding fluid.

4. Are there any limitations to using the divergence theorem?

Yes, the divergence theorem can only be applied to vector fields that are continuous and have a well-defined divergence within the volume enclosed by the closed surface. Additionally, the surface must be closed and the field must be defined on both sides of the surface.

5. Can the divergence theorem be used in higher dimensions?

Yes, the divergence theorem can be generalized to higher dimensions. In three dimensions, it is known as the Gauss's theorem, but in higher dimensions, it is referred to as the generalized Stokes's theorem.

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