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Stokes theorem

  1. Aug 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Hi there. I was trying to solve this problem, from the book. The problem statement says:
    Integrate [tex]\nabla \times{F},F=(3y,-xz,-yz^2)[/tex] over the portion of the surface [tex]2z=x^2+y^2[/tex] under the plane z=2, directly and using Stokes theorem.

    So I started solving the integral directly. For the rotational I got:

    [tex]\nabla \times{F}=\left|\begin{matrix}i&j&k\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\3y&-xz&-yz^{2}\end{matrix}\right|=(x-z^{2},0,-z-3)[/tex]

    Then I parametrized the surface.:

    [itex]\begin{matrix}x=x=r\cos \theta\\ y=r \sin \theta \\z=\displaystyle\frac{r^2}{2}\end{matrix}, \theta[0,2\pi] ,r[0,2][/itex]

    Then I did: [tex]T_r\times{T_{\theta}}[/tex]

    [tex]T_r=(\cos \theta,\sin \theta,r),T_{\theta}=(-r\sin \theta,r\cos \theta,0)[/tex]

    For the cross product I got:
    [tex]T_r\times{T_{\theta}}=(r^2\cos \theta,-r^2\sin \theta,r)[/tex]

    And then I computed the integral

    [tex]\displaystyle\int_{0}^{2}\int_{0}^{2\pi}(r\cos \theta-\displaystyle\frac{r^4}{4},0,\displaystyle\frac{-r^2}{2}-3)\cdot{(-r^2\cos \theta,-r^2\sin \theta,r)}d\theta dr=-12\pi[/tex]

    The result given by the book is: [tex]20\pi[/tex].

    I don't know what I did wrong, and is one of the first exercises that I've solved for the stokes theorem, so maybe I could get some advices and corrections from you :)

    Thank you in advance. Bye.
     
    Last edited: Aug 4, 2011
  2. jcsd
  3. Aug 4, 2011 #2
    I've corrected some typos. Is there anybody out there? :P
     
  4. Aug 5, 2011 #3

    HallsofIvy

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    This is wrong. It should be [itex]\left<-r^2cos(\theta), -r^2sin(\theta), r\right>[/itex] (oriented "upward").

     
  5. Aug 5, 2011 #4
    Thank you very much HallsofIvy. How did you realize that the orientation was negative? I can identify a positive oriented normal in the Cartesian coordinates, but I don't know how to do it in cylindrical or spherical coordinates.
     
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