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Flow through a surface

  1. Aug 23, 2011 #1
    1. The problem statement, all variables and given/known data

    given is the following field [tex] F(x,y,z)=(y,xz,0)[/tex] (F is a vector field) and the borders of the surface are: [tex] 0<x<1 ;y=x ; 0<z<1[/tex] the < should be less equal but I don't know how to do the sign in latex, sorry. The normal vector is given as [tex]n=(a,b,c); b<0[/tex]

    I shall calculate the flow through the surface
    2. Relevant equations

    the formula for the flow, can't type in in latex, sorry, but I think you know which one I mean (flow= integral F*n*dS)


    3. The attempt at a solution
    I need to find my dS, problem here I have, I don't know how to do this exactly. I tried to parametrize it, but I'm not sure how to do it, because I have a function in my borders. So I think y max equals 1 because of the requirements for x. Can anyone help me?
     
  2. jcsd
  3. Aug 23, 2011 #2
    If you try to use [itex]y=x[/itex] anywhere you can, it will be all quite simple.
    Which parametrization did you try ?
     
  4. Aug 23, 2011 #3
    I tried x,x,z not sure though.
     
  5. Aug 23, 2011 #4

    HallsofIvy

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    That surface itself is defined by y= x. The position vector for any point on that surface is r(x, z)= <x, x, z>. Two tangent vectors, at any point are rx= < 1, 1, 0> and another is rz= <0, 0, 1>. The vector differential of surface area is given by [itex]d\vec{S}= \vec{v_s}\times\vec{v_t} dsdt[/itex]. Write your vector function in terms of s and t and integrate the dot product [itex]\vec{F}\cdot d\vec{S}[/itex]
     
    Last edited: Aug 23, 2011
  6. Aug 23, 2011 #5
    Thank you very much, that's what I did after I got the parametrisation. I wasn't sure about it though.
     
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