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Question about divergence theorem and delta dirac function
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- Thread starter Clara Chung
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In summary: In general, if ##f'(x) = g(x)##, then ##f'(x-a) = g(x-a)##. It's a simple application of the chain rule.
- #2
jedishrfu
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I would say by inspection, if you note that for a non-spherical surface the R would in fact be a function: ##R(r,\theta,\phi)## and that ##R(r,\theta,\phi)## factors cancel out using the rules of algebra, leaving an integral of angles only.
Since the ##R(r,\theta,\phi)## factors cancel, then you can choose R to be a spherical surface with no loss in generality.
Perhaps, @Mark44 or @fresh_42 have a more rigorous set of steps.
Since the ##R(r,\theta,\phi)## factors cancel, then you can choose R to be a spherical surface with no loss in generality.
Perhaps, @Mark44 or @fresh_42 have a more rigorous set of steps.
- #3
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Clara Chung said:
You can use the divergence theorem. Take any volume that includes the origin. To avoid the problem at the origin you could remove a small spherical cavity at the origin. The total surface integral is zero and the surface integral of the small cavity is ##-4\pi##.
Last edited:
- #4
- #5
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Clara Chung said:
That's just a substitution ##\vec{r}## to ##\vec{r} - \vec{r'}##.
- #6
Clara Chung
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How do you do the substitution? Why is it ok to let ##\vec{r}## = ##\vec{r} - \vec{r'}## ? They are not equal..PeroK said:
- #7
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Clara Chung said:
In general, if ##f'(x) = g(x)##, then ##f'(x-a) = g(x-a)##. It's a simple application of the chain rule.
Related to Question about divergence theorem and delta dirac function
1. What is the divergence theorem?
The divergence theorem, also known as Gauss's theorem, states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the region enclosed by the surface.
2. How is the divergence theorem related to the delta dirac function?
The divergence theorem can be extended to include the delta dirac function, which is a mathematical tool used to represent a point source or impulse in a vector field. The divergence theorem with the delta dirac function allows for the calculation of the flux through a point source in a closed surface.
3. What is the mathematical expression for the divergence theorem?
The mathematical expression for the divergence theorem is ∫∫∫V (div F) dV = ∫∫S (F · n) dS, where V is the volume enclosed by the closed surface S, F is the vector field, div F is the divergence of F, n is the unit normal vector to the surface, and dV and dS are the volume and surface elements, respectively.
4. How is the divergence theorem used in physics?
The divergence theorem is used in physics to calculate the flow of a vector field through a closed surface, which has applications in fluid dynamics, electromagnetism, and other fields. It is also used in the derivation of important equations, such as the continuity equation and Maxwell's equations.
5. Can the divergence theorem be applied to any vector field?
Yes, the divergence theorem can be applied to any vector field, as long as the field is well-behaved and satisfies certain conditions, such as being continuous and differentiable. However, in some cases, the delta dirac function may need to be used to handle point sources or singularities in the vector field.
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