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September 17th, 2017, 07:09 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3  My First Stokes Theorem problem....
Let F = < z, x, y >. The plane z = 2x + 2y  1 and the paraboloid z = x^2 + y^2 intersect in a closed curve. Stokes's Theorem implies that $\displaystyle \int_{}^{} \int_{D1}^{} ( \nabla ~X~ F ) * N ~dS = \int_{C}^{} F *~dr = \int_{}^{} \int_{D2}^{} ( \nabla ~X~ F ) * N ~DS $ where the line integral is computed over the intersection C of the plane and the paraboloid, and the two surface integrals are computed over the portions of the two surfaces that have boundary C (provided, of course, that the orientations all match). Compute all three integrals Answer: $\displaystyle 3 \pi $ I did the middle line integral successfully, but I would be much appreciated if someone shows me stepbystep of solving $\displaystyle \int_{}^{} \int_{D1}^{} ( \nabla ~X~ F ) * N ~dS $ Last edited by zollen; September 17th, 2017 at 07:11 PM. 
September 17th, 2017, 08:33 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1222 
Let me know if you don't understand this Mathematica sheet Last edited by skipjack; September 19th, 2017 at 08:32 AM. 
September 17th, 2017, 08:40 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1222 
Doing the surface integrals in cylindrical coordinates in this sheet. It's a bit cleaner. 
September 18th, 2017, 04:11 AM  #4 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
Is Wolfram Mathematica home edition free?
Last edited by skipjack; September 19th, 2017 at 08:43 AM. 
September 18th, 2017, 07:44 AM  #5 
Senior Member Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 
I may not have fully understood the syntax. Should you use unit normal vector? Your solutions seem to use (n=Cross(cx,cy)) normal vector.
Last edited by skipjack; September 19th, 2017 at 08:52 AM. 
September 18th, 2017, 09:32 AM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1222  What happens is that the factor that normalizes the unit vector is exactly the same factor that accounts for the local metric of the surface, so the two cancel each other out.
Last edited by skipjack; September 19th, 2017 at 08:53 AM. 
September 19th, 2017, 08:51 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,263 Thanks: 1958  

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