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Homework Help: Outward flux

  1. Jan 15, 2014 #1
    Question: Find the outward flux of the vector field F = i-2j-2k across the surface S defined by z = 4-x2-y2 0≤z≤4

    At first, I used the Divergence Theorem to solve this problem. I took the divF and got the answer of 0. By definition, integrating 0 three times will still equal 0. Thus, the answer I wrote down currently is 0.

    Now there's also another way to find the outward flux, and that is by taking the double integral of Fn

    n = [r_r X r_θ]/[|r_r X r_θ|]
    dσ = |r_r Xr_θ|

    I found r_r X r_θ here from another problem: http://i.imgur.com/UMj72Ub.png

    Checking with WolframAlpha, I got this:
    1st integration: http://www.wolframalpha.com/input/?i=integrate+(-2r^2cos(theta)+4r^2sin(theta)-2r)+dr+from+0+to+2

    2nd integration: http://www.wolframalpha.com/input/?i=integrate+(-4/3)(-8sin(theta)+4cos(theta)+3)+from+0+to+2pi

    I would greatly appreciate it if someone can explain why the answer using the Divergence theorem is not equal to the Outward Flux when taking the double integral.


    Last edited: Jan 15, 2014
  2. jcsd
  3. Jan 15, 2014 #2
    Never mind, figured out my mistake. I should've not used the parametrized form to take the double integral. I fixed this by making n = [-2x-2y]/|-2x-2y| and dσ = |-2x-2y| (I acquired these values from finding the gradient of z).
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