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Simple vector calculus

  1. Nov 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose we have the vector field F whose x component is given by [itex]F_{x}=Ax[/itex] and whose divergence is known to be zero [itex] \vec{\nabla}\cdot\vec{F}=0[/itex], then find a possible y component for this field. How many y components are possible?

    2. The attempt at a solution

    So the divergence in cartesian coordinates is given by
    [tex]\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} = 0[/tex]

    Using the fact that [tex]F_{x}=Ax[/tex]
    [tex]A+\frac{\partial F}{\partial y} = 0[/tex]
    [tex]\frac{\partial F}{\partial y} = -A[/tex]
    integrate both sides with respect to y we get

    [tex]F_{y}=-Ay+B[/tex]

    where B is a constant
    is that sufficient for a possible y component? For the question with howm any are possible... arent there infinite possibilities since B could be anything. But they are all parallel to each... linearly dependant on the above answer?
     
  2. jcsd
  3. Nov 9, 2008 #2
    As far as I'm concerned, you're good to go.
     
  4. Nov 9, 2008 #3
    but.. infinitely many solutions becuase of B or finite becuase they are all linearly dependant on the solution given?
     
  5. Nov 10, 2008 #4
    There is an infinite amount of parallel solutions.
     
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