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Clara Chung said:View attachment 238665
How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
PeroK said:You can use the divergence theorem. Take any volume that does not include 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##.
Clara Chung said:View attachment 238678
How to get 1.100 from 1.99? I can't find the derivation in the book...
How do you do the substitution? Why is it ok to let ##\vec{r}## = ##\vec{r} - \vec{r'}## ? They are not equal..PeroK said:That's just a substitution ##\vec{r}## to ##\vec{r} - \vec{r'}##.
Clara Chung said:How do you do the substitution? Why is it ok to let ##\vec{r}## = ##\vec{r} - \vec{r'}## ? They are not equal..
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.
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.
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.
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.
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.