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Homework Help: Curl div

  1. Jun 14, 2010 #1
    Curl div....

    1. The problem statement, all variables and given/known data

    f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for?

    I need to know if a scalar field can have the meanings of roration and diverge like a vector field
     
  2. jcsd
  3. Jun 14, 2010 #2

    lanedance

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    Homework Helper

    Re: Curl div....

    if f is a scalar field, then grad(f) is a vector fields

    div(f) makes no sense as f is a scalar, and div operates on vectors , curl(f) doesn't make sense for the same reasons

    i'm guess rotgrad(f) = curl(grad(f)) which is ok, though i remember correctly its zero

    and divgrad(f) = div(grad(f)) which is ok as well

    have a look at this
    http://en.wikipedia.org/wiki/Vector_calculus_identities
     
    Last edited: Jun 14, 2010
  4. Jun 14, 2010 #3
    Re: Curl div....

    div and curl are only defined for vector fields. grad is only defined for scalar fields.

    The result of div is a scalar and the result of grad and curl is a vector. Therefore, these are the second spatial derivatives that you can construct:

    [tex]
    \mathrm{div} (\mathbf{grad} \, \phi) = \nabla^{2} \, \phi
    [/tex]

    [tex]
    \mathrm{div} (\mathbf{curl} \, \mathbf{A}) = 0
    [/tex]

    [tex]
    \mathbf{grad} (\mathrm{div} \, \mathbf{A})
    [/tex]

    [tex]
    \mathbf{curl}(\mathbf{grad} \, \phi) = \mathbf{0}
    [/tex]
    [tex]
    \mathbf{curl} (\mathbf{curl} \, \mathbf{A}) = \mathbf{grad} (\mathrm{div} \, \mathbf{A}) - \nabla^{2} \, \mathbf{A}
    [/tex]

    where [itex]\nabla^{2}[/itex] stands for the Laplace differential operator (Laplacian).
     
  5. Jun 14, 2010 #4

    lanedance

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    Homework Helper

    Re: Curl div....

    fixed up above - missed the curl(f) bit
     
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