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Question about del

  1. Jan 30, 2012 #1
    Question about "del"

    We know that A x (BxC)= (A·C)B-(A·B)C (*)

    In the following example, we can treat ∇ as a vector and apply the formula (*) above to get the correct answer
    ∇x(∇xV)= ∇(∇·V)-∇^2 V

    But in this example, the formula (*) seems to fail
    ∇x(UxV)≠U(∇·V)-V(∇·U)

    Why?
     
  2. jcsd
  3. Jan 30, 2012 #2

    Char. Limit

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    Re: Question about "del"

    Because ∇ is NOT a vector, no matter how much we want it to act like one. For one, ∇ isn't commutative. (Vectors are)
     
  4. Jan 30, 2012 #3
    Re: Question about "del"

    Then why the textbook simply uses the formula (*) when deriving ∇x(∇xV)= ∇(∇·V)-∇^2 V ? Is it just a coincidence that the formula (*) works for ∇x(∇xV)= ∇(∇·V)-∇^2 V ?
     
  5. Jan 31, 2012 #4

    dextercioby

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    Re: Question about "del"

    It's just a coincidence. Nabla is a differential operator. You can't simply switch it from one term of a an equation to another without changing the result.

    Vector identities are always easier to express in tensor notation, using delta Kronecker and epsilon Levi Civita.
     
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