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Flux and Circulation

  1. Aug 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Let F(x,y) = ( P(x,y), Q(x,y)) be a vector field that is continuously differentiable along the closed smooth curve C : x2+y2 = 1. Moreover let

    -F(x,y) = F( -x, -y)≠ 0 and

    P(x,y) = -P(-x,y) and Q(x,y) = Q(-x,y).

    Determine all the possible values of the circulation around C, and argue why flux across C is non zero.

    2. Relevant equations

    Circulation = ∫CF. T ds , T tangent vector
    Flux = ∫CF. N ds , N normal vector

    3. The attempt at a solution

    I'm not quite sure where to begin . Do we parametrize C by r(t) = (cos t, sin t) 0≤ t≤ 2[itex]\pi[/itex], and then use the definitions of flux and circulation?
     
  2. jcsd
  3. Aug 26, 2012 #2
    From the given relations among F, P and Q you can deduce how they are related in all four quadrants. What will that mean with regard to the integrals?
     
  4. Aug 26, 2012 #3
    Ah now I see for the flux part;

    Look at each quadrant {x,y>0} {x,y<0} {x>0, y<0} and {x<0,y>0}

    When x, y> 0, The integral evaluates to a value.

    Let ∫CF.n ds = f

    Then for x,y<0

    the integral becomes ∫F( -P, -Q) .n ds = - f

    Then for x> 0, y< 0
    ∫F(P(x,-y), Q(x,-y)).n ds= a ≠f


    Then for x< 0, y> 0

    ∫F(P(-x,y),Q(-x,y)).n ds =∫ F( -P(x,y),Q(x,y)).n ds =b ≠a.

    So you can see the total flux is non zero. The two regions x,y>0 and x,y<0 cancel out
    and the other two are not equal.
     
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