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Homework Help: Vector calculus, identities

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that v[tex]\nabla[/tex]v = [tex]\nabla[/tex]xvxv
    v · ∇v = ∇(0.5v2 + c × v

    c=∇ × v

    My attempt
    ∇(A · B)= B · ∇A + A · ∇B + B×(∇×A) + A×(∇×B)

    Replace A and B with V

    ∇(v · v)= v · ∇v + v · ∇v + v×(∇×v) + v×(∇×v)

    v · ∇v = ∇(0.5v2 - v×(∇×v)

    Is v×(∇×v) = =∇ × v × v?
    And am I on the right track?

    I can't fint such a rule in my textbook
     
  2. jcsd
  3. Apr 28, 2010 #2

    lanedance

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    doesn't vxv = 0?
     
  4. Apr 28, 2010 #3

    gabbagabbahey

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    What you've written doesn't make much sense....Do you mean [itex]v\matbf{\nabla}v=(\mathbf{\nabla}\times \textbf{v})\times\textbf{v}+(\textbf{v}\cdot\mathbf{\nabla})\textbf{v}[/itex] ?


    Well, [itex]\textbf{a}\times\textbf{b}=-\textbf{b}\times\textbf{a}[/itex], so you tell us whether or not [itex](\mathbf{\nabla}\times \textbf{v})\times\textbf{v}=\textbf{v}\times(\mathbf{\nabla}\times \textbf{v})[/itex]:wink:
     
    Last edited: Apr 28, 2010
  5. Apr 28, 2010 #4
    Thanks to both of you.
    The first line is wrong

    I was suppose to show that
    v · [tex]\nabla[/tex]v = ∇[tex]\nabla[/tex](0.5v2) + c × v

    c=∇[tex]\nabla[/tex]× v.
    I then usend
    ∇[tex]\nabla[/tex](A · B)=B · [tex]\nabla[/tex]A + A · [tex]\nabla[/tex]B + B×([tex]\nabla[/tex]×A) + A×([tex]\nabla[/tex]×B)

    Replaced A and B with v

    And got

    v[tex]\nabla[/tex]v = [tex]\nabla[/tex](0.5v2) - v×([tex]\nabla[/tex]×v)

    Using A×B = - B×A

    - v×([tex]\nabla[/tex]×v) = ([tex]\nabla[/tex]×vv

    And I have:

    v · [tex]\nabla[/tex]v = ∇[tex]\nabla[/tex](0.5v2) + c × v

    c=∇[tex]\nabla[/tex]× v.

    Am I right?

    Thanks again for all the help
     
  6. Apr 28, 2010 #5

    gabbagabbahey

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    Looks good to me:smile:
     
  7. Apr 28, 2010 #6
    Thanks.
     
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