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If the divergence of a vector field is zero

  1. Feb 10, 2007 #1
    1. The problem statement, all variables and given/known data
    If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector?

    2. Relevant equations
    Just the equations for divergence and curl. In TeX:
    [tex]\nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}[/tex]
    and the equivalent for curl.

    3. The attempt at a solution
    I really don't know at all how to find an answer.
    Last edited by a moderator: Feb 11, 2007
  2. jcsd
  3. Feb 11, 2007 #2
    The divergence of the curl of ANY vector is =0. You cant find that "vector" without some more information, eg boundary conditions.
  4. Feb 11, 2007 #3


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  5. Feb 11, 2007 #4
    So when a problem gives a vector field where it's divergence is zero, and it asks to find a vector field such that the curl of the vector field is the given vector field, I can just choose any vector field?
  6. Feb 11, 2007 #5


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    No, those responses were to what you had posted before- that all you knew about the vector field was that its divergence was equal to 0. You did not say you were given a vector field that happened to have divergence equal to 0!

    If you are given a vector field, say, u(x,y,z)i+ v(x,y,z)j+ w(x,y,z)k with divergence 0, Then write out the formula for curl of a vector field and set the components equal:
    [tex]\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z}= u[/tex]
    [tex]\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}= v[/tex]
    [tex]\frac{\partial g}{\partial x}- \frac{\partial f}{\partial x}= w[/tex]

    Solve those for f, g, h,
  7. Feb 11, 2007 #6
    I already knew that; I suppose I just didn't write it out clearly enough. But what was confusing me was how to solve for those. It seems like that is a system of PDEs, and I have no idea how to solve those.
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