Verify Divergence Theorem for V = xy i − y^2 j + z k and Enclosed Surface

[nestleeng]

Homework Statement

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.

Homework Equations

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[PLAIN]https://upload.wikimedia.org/math/7/b/7/7b759968274f2f43cfaab3ce5672da74.png[PLAIN]https://upload.wikimedia.org/math/a/b/9/ab9fd5a4aaa36e402c98cbd36af3a70d.png

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

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.

 

[HallsofIvy]

You haven't really said much about what you did. What did you get for $\nabla\cdot\vec{V}$? What are s and a? Constants? Then "(ii) s= 1, $0\le z\le 1$" makes no sense.

 

[vela]

Please read https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686784/, paying particular attention to items 3 and 4.