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Verify Divergence Theorem

  1. Jan 17, 2016 #1
    1. The problem statement, all variables and given/known data
    Verify the divergence theorem for the function
    V = xy i − y^2 j + z k
    and the surface enclosed by the three parts
    (i) z = 0, s < 1, s^2 = x^2 + y^2,
    (ii) s = 1, 0 ≤ z ≤ 1 and
    (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1.

    2. Relevant equations

    7ec662d9708da6cde59c16ccc768f4bf.png 25px-OiintLaTeX.svg.png [PLAIN]https://upload.wikimedia.org/math/7/b/7/7b759968274f2f43cfaab3ce5672da74.png[PLAIN]https://upload.wikimedia.org/math/a/b/9/ab9fd5a4aaa36e402c98cbd36af3a70d.png [Broken]

    Divergence theorem, although on the RHS I put vector DS = nDS.

    3. The attempt at a solution
    So I solved the LHS and got the answer to be a*Pi

    on the RHS, splitting the 3 surfaces,
    (i) got 0 for integral
    (ii) got 0 for integral
    (iii) staying in cartesians, I have to integrate ((1-a^2)(-x^2 y +y^3 + x^2 +y^2) +a^2)/Sqrt(a^2+(1-a^2)(x^2+y^2) dxdy between -1 and 1 for x and y which even Wolfram can't do.

    Spent hours on this please help.
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Jan 17, 2016 #2

    HallsofIvy

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    You haven't really said much about what you did. What did you get for [itex]\nabla\cdot\vec{V}[/itex]? What are s and a? Constants? Then "(ii) s= 1, [itex]0\le z\le 1[/itex]" makes no sense.
     
  4. Jan 17, 2016 #3

    vela

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