Help with using the Divergence Theorem

In summary, the divergent theorem states that the surface integral of a function over a solid surface converges to zero as the distance from the point of interest increases.
  • #1
Divergent13
48
0
Hi!

We are nearing the end of our course --- culminating in Stokes and Divergence Theorems for surface integrals, and I am having some difficulty with the following

1. F(x,y,z) = [tex]<x^3y, -x^2y^2, -x^2yz>[/tex]

where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z = -2 and z=2.

I computed div F properly...

Well I know what the z limits are in my Triple Integral, however what must I use as my radius? Theta should go from 0 to 2pi correct?

2. F(x,y,z) = [tex]<x^2y, xy^2, 2xyz>[/tex]

where S is the surface of the tetrahedron bounded by the planes x=0, y=0, z=0, and x+2y+z = 2

Here must my triple integral be from 0 to 2 for the x limit, then 0 to (2-x)/2 for my y limits, and for z just 0 to 2-x-2y.

Those seem correct, but a confirmation would be nice! Thanks a lot!
 
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  • #2
Whoops -- it's probably implied but the questions ask to compute


[tex]\int\int_SFdS[/tex] = [tex]\int\int\int_EdivFdV[/tex]
 
Last edited:
  • #3
1. Did you get the divergence to be zero?
I did, so is this an exercise in verifying that computing the surface integral directly also yields zero?
 
  • #4
The limits in 2 seems right.
 
  • #5
Ahh so maybe i didnt compute div F properly-- yup it's 0, making the answer 0. Thanks.
 
  • #6
Ahh here's another one that's a bit more challenging:


Its BEGGING spherical coordinates:

[tex]F(x,y,z) = <x^3+ysinz, y^3+zsinx, 3z>[/tex]

S: Surface of the solid bounded by the hemispheres [tex]z=sqrt(4-x^2-y^2)[/tex] and [tex]z=sqrt(1-x^2-y^2)[/tex] and the plane [tex]z=0[/tex].

I set: r(r,phi,theta) = <rsin(phi)cos(theta), rsin(phi)sin(theta), rcos(phi) >
Are the following the correct limits of integration?

1 < r < 2
0 < phi < pi/2
0< theta < 2pi
 
  • #7
Are you talking about the limits of integration for the surface here??
(Remember, if you're talking about the solid, r=0 is certainly included.)

In order to solve this, I suggest you split your region in two solids, the two hemishperes, both bounded by the plane z=0.

Then it is easy to apply spherical coordinates on these solids separetely.
 
  • #8
I'm sorry, total mistak on my parte (I thought there was a minus sign somewhere :eek:)
Your approach is perfefctly correct
 
  • #9
So it is from 1 to 2... okay thanks!
 

Related to Help with using the Divergence Theorem

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the flow of a vector field through a closed surface to the behavior of the vector field inside the surface.

2. How is the Divergence Theorem used in science?

The Divergence Theorem is a powerful tool in physics and engineering, as it allows for the calculation of flux, which is the amount of a vector field passing through a surface. This is useful in understanding and solving problems related to fluid dynamics, electromagnetism, and other fields.

3. What are some real-life applications of the Divergence Theorem?

The Divergence Theorem has many practical applications, such as calculating fluid flow through a pipe, determining the electric field around a charged object, and analyzing the flow of heat in a system.

4. What are the requirements for using the Divergence Theorem?

To use the Divergence Theorem, the vector field must be defined and continuous inside the surface, and the surface must be closed and bounded.

5. How can I apply the Divergence Theorem in my own research or experiments?

If your research or experiments involve vector fields and closed surfaces, the Divergence Theorem can be a useful tool for calculating flux and understanding the behavior of the vector field. It is important to ensure that the requirements for using the theorem are met in your specific case.

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