Solving Gauss Divergence Theorem on a Closed Surface

In summary, the homework statement is trying to solve for the left hand side which appear to be 972*pi)/5, however they can't seems to solve the right hand side. They substitute x = 3sin(theta)cos(phi), y=3sin(theta)sin(phi), z=3cos(theta) and get N=9sin^2(theta)cos(phi)i+9sin^2(theta)cos(phi)j+9cos(theta)sin(θ)+27sin^3(θ)cos^2(φ)+27sin^2(θ)cos(θ)sin^2(φ),
  • #1
HeheZz
11
0

Homework Statement



Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA
Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk


The Attempt at a Solution



I have tried to solve the left hand side which appear to be (972*pi)/5
However, I can't seems to solve the right hand side to get the same answer.
I substitute x = 3sin(theta)cos(phi), y=3sin(theta)sin(phi), z=3cos(theta)
Therefore N=9sin^2(theta)cos(phi)i+9sin^2(theta)cos(phi)j+9cos(theta)
and F = F=27sin(θ)cos^2(θ)cos(φ)+27sin^3(θ)cos^2(φ)sin(φ)+27sin^2(θ)cos(θ)sin^2(φ)
then I used ∫(0-2pi)∫(0-pi) F. (N) dθdφ
I got the final answer as (324*pi)/5 which does not match with left hand side.
Hope anyone can help here please. Thanks!
 
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  • #2
Your N is wrong. Describe to me what you think N is.
 
  • #3
N is the normal vector?

where r(θ,φ) = (3sinθcosφ, 3sinθsinφ, 3cosθ)

and N = rθ X rφ

Thus, N=9sin^2(θ)cos(φ)i+9sin^2(θ)cos(φ)j+9cos(θ)sin(θ)k
 
  • #4
Yes, it's a normal vector. More important, it's the unit normal vector. Since you're using a sphere, it will just be the radial unit vector.

Edit: Oh, I see what you're doing. That's not just the normal vector but the normal vector scaled by the part of the area element. I think you're just cranking out the integral wrong, but let me try calculate it here to make sure.
 
Last edited:
  • #5
Do you mean the range values when i am integrating? I don't understand what to use. Isnt it 0-pi for the inner integral and 0-2pi for the outer integral?
 
  • #6
HeheZz said:
N is the normal vector?

where r(θ,φ) = (3sinθcosφ, 3sinθsinφ, 3cosθ)

and N = rθ X rφ

Thus, N=9sin^2(θ)cos(φ)i+9sin^2(θ)cos(φ)j+9cos(θ)sin(θ)k
The y-component of your N should have sin(φ), not cos(φ).
 
  • #7
HeheZz said:
Do you mean the range values when i am integrating? I don't understand what to use. Isnt it 0-pi for the inner integral and 0-2pi for the outer integral?
Yeah, you did everything right except you made a mistake on the y-component of N. If you fix that, you should get the right answer. It worked out for me.
 
  • #8
OK! I got it! Tnx for the help very much! I didnt realize this mistake and was really stress over it.. Thanks again for the help and I got the answer :D
 

1. What is the Gauss Divergence Theorem?

The Gauss Divergence Theorem, also known as the Gauss-Ostrogradsky Theorem, is a fundamental theorem in vector calculus that relates a surface integral over a closed surface to a triple integral over the enclosed volume. It is named after mathematicians Carl Friedrich Gauss and Mikhail Vasilevich Ostrogradsky.

2. What is the significance of the Gauss Divergence Theorem?

The Gauss Divergence Theorem is significant because it allows for the simplification of complicated integrals involving vector fields. It also has applications in several areas of physics, such as fluid dynamics, electromagnetism, and thermodynamics.

3. How is the Gauss Divergence Theorem derived?

The Gauss Divergence Theorem can be derived using the Divergence Theorem, which relates a surface integral to a volume integral over a region in space. By applying the Divergence Theorem to a vector field, we can then extend it to a closed surface and obtain the Gauss Divergence Theorem.

4. What are the conditions for the Gauss Divergence Theorem to hold?

The Gauss Divergence Theorem holds if the vector field is continuous and has continuous first-order partial derivatives over the region of interest. The surface must also be smooth and closed, meaning it has no boundary.

5. What are some real-world applications of the Gauss Divergence Theorem?

The Gauss Divergence Theorem has numerous applications in physics and engineering, such as calculating fluid flow rates, determining the electric field inside a charged closed surface, and analyzing heat transfer in thermodynamics. It is also used in computer graphics for 3D modeling and animation.

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