Help with using the Divergence Theorem

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Discussion Overview

The discussion revolves around the application of the Divergence Theorem in computing surface integrals and triple integrals for specific vector fields and bounded regions. Participants are working through examples involving different geometries, including hyperboloids and tetrahedrons, and are seeking confirmation on their limits of integration and calculations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents a vector field F and asks about the limits of integration for a solid bounded by a hyperboloid and planes, seeking confirmation on their calculations.
  • Another participant notes that the divergence of the first vector field is zero, questioning if the exercise is to verify that the surface integral also yields zero.
  • A different vector field is introduced, with a participant providing limits of integration for a tetrahedron and asking for validation of their approach.
  • One participant suggests splitting the region into two solids for easier application of spherical coordinates when discussing a more complex vector field bounded by hemispheres.
  • Another participant acknowledges a mistake in their understanding and confirms the correctness of the approach taken by others regarding limits of integration.

Areas of Agreement / Disagreement

There is some agreement on the limits of integration for the second vector field, but uncertainty remains regarding the calculations for the first vector field and the divergence. Participants express differing levels of confidence in their computations and interpretations.

Contextual Notes

Participants express uncertainty about the divergence calculations and the correct limits of integration, indicating potential dependencies on their interpretations of the problems and the geometries involved.

Who May Find This Useful

Students and individuals studying vector calculus, particularly those interested in the Divergence Theorem and its applications in various geometrical contexts.

Divergent13
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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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Whoops -- it's probably implied but the questions ask to compute


[tex]\int\int_SFdS[/tex] = [tex]\int\int\int_EdivFdV[/tex]
 
Last edited:
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?
 
The limits in 2 seems right.
 
Ahh so maybe i didnt compute div F properly-- yup it's 0, making the answer 0. Thanks.
 
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
 
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.
 
I'm sorry, total mistak on my parte (I thought there was a minus sign somewhere :eek:)
Your approach is perfefctly correct
 
So it is from 1 to 2... okay thanks!
 

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