Verifying divergence theorem with an example

In summary, to verify the divergence theorem when F=xi+yj+zk and sigma is the closed surface bounded by the cylindrical surface x^2+y^2=1 and the planes z=0, z=1, you need to break up the cylindrical surface into three parts (top, bottom, and sides), perform the integral of F over each surface, and sum them up. The triple integral side of the equation should result in 3pi, but the method for solving the flux side of the equation \oint\ointF.ds is not specified. Any further help would be appreciated.
  • #1
grissom
2
0
Verify the divergence theorem when F=xi+yj+zk and sigma is the closed surface bounded by the cylindrical surface x^2+y^2=1 and the planes z=0, z=1.

I've done the triple integral side of the equation and got 3pi but don't know how to solve the flux side of the equation [tex]\oint\oint[/tex]F.ds.

Any help is appreciated. Also, this is my first time using this, so the symbols may be a bit off.
 
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  • #2
You have to break up the cylindrical surface into three parts: the top, bottom and the "sides" for the lack of a better word. Then, perform the integral of F over each surface and finally, sum them up.
 

Related to Verifying divergence theorem with an example

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical principle that relates the flow of a vector field through a closed surface to the behavior of the field inside the surface. It states that the total outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the region enclosed by the surface.

2. How is the divergence theorem verified?

The divergence theorem can be verified by using a specific example, where the vector field and the closed surface are known. The surface integral of the normal component of the vector field over the surface is calculated, and the volume integral of the divergence of the field over the region enclosed by the surface is also calculated. If these two values are found to be equal, then the divergence theorem is verified.

3. Can you provide an example of verifying the divergence theorem?

For example, let's consider a vector field F(x,y,z) = (x, y, z) and a closed surface S defined by the cylinder x^2 + y^2 = 1, 0 ≤ z ≤ 2. We can calculate the surface integral of the normal component of F over S by using the formula ∫∫F⋅n dS, where n is the unit normal vector to the surface. This gives us a value of 4π for the surface integral. Next, we can calculate the volume integral of the divergence of F over the region enclosed by S by using the formula ∫∫∫∇⋅F dV. This gives us a value of 4π for the volume integral. Since the two values are equal, we can conclude that the divergence theorem is verified for this example.

4. What are the applications of the divergence theorem?

The divergence theorem has various applications in mathematics and physics. It is used to solve problems involving fluid flow, electric and magnetic fields, and heat conduction. It also has applications in differential equations, geometric modeling, and computer graphics.

5. Is the divergence theorem only applicable to three-dimensional spaces?

No, the divergence theorem can be extended to any number of dimensions. In three dimensions, it is known as the 3D divergence theorem or Gauss's theorem. In two dimensions, it is known as the 2D divergence theorem or Green's theorem. In one dimension, it is known as the 1D divergence theorem or the fundamental theorem of calculus.

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