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Converting surface integral to line

  1. Jan 15, 2014 #1
    1. The problem statement, all variables and given/known data
    I have the following integral
    [tex]
    \int_{ABC}{\mathbf{v}\cdot \nabla f_id\sigma}
    [/tex]
    where $d\sigma$ is an area element, $\mathbf v$ is a velocity vector and [itex]f_i[/itex] some function. The integral is performed across a triangle ABC and it is assumed that f is linear.

    In my book this integral becomes
    [tex]
    \mathbf v\cdot \int_{AB}{f_id\mathbf l} + I_s,
    [/tex]
    where [itex]I_s[/itex] is the flux across [itex]BC[/itex] and [itex]AC[/itex]. Can someone explain to me how this integral is solved?
     
  2. jcsd
  3. Jan 15, 2014 #2

    HallsofIvy

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    Staff Emeritus
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    Use "Stoke's theorem" that you should remember from Calculus:
    [tex]\oint_C \vec{f}\cdot d\vec{r}= \int_\sigma\int \nabla\times\vec{f}dS[/tex]
    where C is the boundary of surface [itex]\sigma[/itex].
     
  4. Jan 15, 2014 #3
    Hi HallsofIvy

    Thanks, I remember that. But the RHS is in terms of a rotation operatir, not the gradient as I have.
     
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