Calculating Flux through a Sphere using Divergence Theorem

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Homework Help Overview

The discussion revolves around calculating the flux through a sphere using the Divergence Theorem, specifically focusing on evaluating a vector field's divergence and integrating over a spherical volume.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to compute the divergence of a vector field and set up a triple integral in spherical coordinates. Some participants suggest converting terms collectively to simplify the integration process. Others question the limits of integration, particularly the angle phi, which the original poster initially set incorrectly.

Discussion Status

Participants are actively engaging with the problem, offering suggestions for simplifying the integral and questioning the setup. There is an acknowledgment of a potential error in the limits of integration, indicating a productive direction in the discussion.

Contextual Notes

The original poster expresses frustration over obtaining an unexpectedly large answer, suggesting there may be a misunderstanding or oversight in the setup or calculations. The problem context includes a specific sphere radius and a vector field defined at the origin.

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Evaluate [URL]http://webwork.latech.edu/webwork2_files/tmp/equations/93/91cfe28c766cad38444f0213c651281.png[/URL] where [URL]http://webwork.latech.edu/webwork2_files/tmp/equations/59/a56001472f977192637ea927c607a61.png[/URL] and is the surface of the sphere of radius 6 centered at the origin.

Ok so I started by taking the divF to get 3y^2+3x^2+3z^2. Using polar coordinates I created the integral (from 0 to 2pi) of the integral (from 0 to pi/2) of the integral (from 0 to 6) (3r2sin2(theta)sin2(phi)+3r2cos2(theta)sin2(phi)+3r2cos2(phi)))r2sin(phi) d(phi)d(theta)d(r).

After all of that I simplified my answer down but I keep getting a huge number, 29314.82937, for an answer and it's incorrect. This problem seems very easy so I feel like I'm missing a very obvious step.
 
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If it helps, instead of converting each term, convert them as a group to r^2.
 
That is a good idea, but I still didn't come out with the right answer.
 
Why are you taking phi from 0 to pi/2? It should be to pi.
 
That is the million dollar question. Thank you very much my friend.
 

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