- #1
Question about divergence theorem and delta dirac function
- Context: Undergrad
- Thread starter Clara Chung
- Start date
Click For Summary
SUMMARY
The discussion centers on proving the validity of the divergence theorem in relation to the line integral equating to 4π for any closed surface containing the origin. Participants emphasize the cancellation of the R(r, θ, φ) factors, allowing for the simplification to a spherical surface without loss of generality. The divergence theorem is applied to demonstrate that the total surface integral remains zero when excluding a small spherical cavity at the origin, confirming the integral's value as -4π. Substitution techniques are also discussed, specifically the transformation of vector variables using the chain rule.
PREREQUISITES- Understanding of the divergence theorem
- Familiarity with vector calculus and surface integrals
- Knowledge of spherical coordinates (r, θ, φ)
- Proficiency in applying the chain rule in calculus
- Study the application of the divergence theorem in various contexts
- Explore vector calculus textbooks for detailed examples of surface integrals
- Investigate spherical coordinates and their applications in physics
- Review substitution methods in calculus, particularly in vector fields
Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector calculus and the divergence theorem, particularly in relation to surface integrals and transformations.
- #2
jedishrfu
Mentor
- 15,681
- 10,476
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
- 29,619
- 21,427
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
- 29,619
- 21,427
Clara Chung said:
That's just a substitution ##\vec{r}## to ##\vec{r} - \vec{r'}##.
- #6
Clara Chung
- 300
- 13
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
- 29,619
- 21,427
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.
Similar threads
- · Replies 25 ·
- · Replies 3 ·
Undergrad
Divergence theorem and closed surfaces
- · Replies 2 ·
Undergrad
Vector field and Helmholtz Theorem
- · Replies 20 ·
- · Replies 2 ·
- · Replies 6 ·
- · Replies 9 ·
- · Replies 4 ·
Undergrad
Question about the Dirac delta function
- · Replies 8 ·