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Divergence and F=∇×A

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    A vector field F for which div F = 0, is called incompressible (also called solenoidal). Consider the vector field F(x, y, z) = ⟨y, x + y, −z⟩.

    (a) (1 point) Show that F is incompressible.

    (b) (3 points) Find a vector field A such that F=[tex]\nabla[/tex]×A.

    2. Relevant equations
    div F = [tex]\nabla[/tex] . F

    3. The attempt at a solution

    I understand how to do part a and confirmed that div F = 0 and thus is incompressible.
    But I'm not entirely sure how to find part b.

    Is there a cross product operation in which A = some combination of F and [tex]\nabla[/tex]?

    Or would you assign A some arbitrary vectors [tex]\left\langle[/tex]P,Q,R⟩ and take the cross of those with [tex]\nabla[/tex]. Resulting in the vector
    ⟨y, x + y, −z⟩ = (dR/dy-dQ/dz)i-(dR/dx-dP/dz)j+(dQ/dx-dP/dy)k
    and somehow solve for P,Q,R?

    Or am I completely on the wrong track?

    Any help or advice would be wonderful!
  2. jcsd
  3. Mar 9, 2010 #2


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    You can find P, Q, and R by trial and error pretty quickly since F is relatively simple.
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