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How can I find this surface integral in cylindrical coordina

  1. Feb 5, 2017 #1
    1. The problem statement, all variables and given/known data
    A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by

    $\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda z)\hat k $

    where $\hat i$ , $\hat j$ , and $\hat k$ are the unit vectors along the cartesian axes and $\hat \rho$ is the unit vector $\frac{x}{\rho}\hat i \ + \frac{y}{\rho}\hat j$.

    (a) Calculate, as a surface integral, the flux of $\vec F$ through the closed surface bounded by the cylinders $\rho = a$ and $\rho = 2a$ and the planes $z =\pm \frac{a\pi}{2} $.

    (b) Evaluate the same integral using the divergence theorem.

    So I am pretty lost on how to even begin part (a). I sketched a picture of the situation and am I kinda confused as to how to represent the unit normal vector, as well as the infinitesimal area elements. Am I supposed to take four separate integrals and sum them afterwards, one for the inner and outer curved surfaces, and one for the upper and lower caps?
     
  2. jcsd
  3. Feb 6, 2017 #2

    LCKurtz

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    I see that this is marked solved, so I haven't commented. I just fixed the text to make it readable. @John004: use ## instead of $ to delimit inline Latex.

     
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