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Something about vector fields

  1. Jun 16, 2011 #1
    1. The problem statement, all variables and given/known data
    This problem is in Introduction to Eletrodynamics, of Griffiths, 3rd edition, p.20, problem 1.19. He asks a vector function v(x,y,z), other than the constant, that has:
    [itex]\nabla\cdot\vec{v}=0 \mbox{ and } \nabla\times\vec{v}=0[/itex]


    2. Relevant equations
    I hope you know them.


    3. The attempt at a solution
    I tried to force the divergence to be zero, using [itex]\vec{v}[/itex], like this: [itex]\vec{v}=v_x(y,z)\hat{x}+v_y(x,z)\hat{y}+v_z(x,y) \hat{z}[/itex]
    then i solved for the curl of v to be zero and this gave me 3 partial diferencial equations, and so I stoped and decided to get help. Some ideas?
     
  2. jcsd
  3. Jun 16, 2011 #2

    Pengwuino

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    One convenient fact that probably will help you is the fact that [itex]\nabla \times (\nabla f(\vec{x})) = 0[/itex].

    Let [itex]\vec{v} = \nabla f(\vec{x})[/itex] and your second requirement is automatically satisfied. Then you just need to determine what 'f' is based on the first requirement.
     
  4. Jun 16, 2011 #3

    LCKurtz

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    Another suggestion -- look for a 2D field F = <u(x,y),v(x,y),0> and think about the Cauchy Riemann equations.
     
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