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Solenoidal proof

  1. Mar 1, 2012 #1
    Be f and g two differentiable scalar field. Proof that ([itex]\nabla[/itex]f) x ([itex]\nabla[/itex]g) is solenoidal.
     
  2. jcsd
  3. Mar 1, 2012 #2

    Dick

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    Show what you've done so far. What would you do to show a vector field is solenoidal?
     
  4. Mar 1, 2012 #3
    Well, to be solenoidal, I know that [itex]\nabla[/itex] [itex]\cdot[/itex] ([itex]\nabla[/itex]f x [itex]\nabla[/itex]g) needs to be 0.

    So,

    [itex]\nabla[/itex] [itex]\cdot[/itex] ([itex]\nabla[/itex]f x [itex]\nabla[/itex]g) = [itex]\nabla[/itex]g [itex]\cdot[/itex] ([itex]\nabla[/itex] x [itex]\nabla[/itex]f) - [itex]\nabla[/itex]f [itex]\cdot[/itex] ([itex]\nabla[/itex] x [itex]\nabla[/itex]g)

    Right? But why is this equal to zero?
     
  5. Mar 1, 2012 #4

    I like Serena

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    Last edited: Mar 1, 2012
  6. Mar 2, 2012 #5
    Thank you so much!
     
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