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Curl Product Rule confusion?

  1. Dec 28, 2008 #1
    Curl Product Rule confusion???

    1. The problem statement, all variables and given/known data
    In Griffith's Introduction to Electrodynamics, he gives the rule:

    [tex]\nabla\times(\bold{A}\times\bold{B})=(\bold{B}\cdot\nabla)\bold{A}-(\bold{A}\cdot\nabla)\bold{B}+\bold{A}(\nabla\cdot\bold{B})-\bold{B}(\nabla\cdot\bold{A})[/tex]

    Now I know I am missing something stupid here, but what is the difference between [itex](\bold{A}\cdot\nabla)\bold{B}[/itex] and [itex]\bold{B}(\nabla\cdot\bold{A})[/itex] ?


    The dot product commutes doesn't it? And then we are left with a scalar times a vector

    If [itex](\bold{A}\cdot\nabla)=(\nabla\cdot\bold{A})=k[/itex]

    then what is the difference between kB and Bk ?


    I know I am doing something wrong, but what?

    Casey
     
  2. jcsd
  3. Dec 28, 2008 #2
    Re: Curl Product Rule confusion???

    Hi Saladsamurai,

    The dot product on a vector space commutes because the scalars commute. The dot product in your formula isn't quite the same. Notice that, for example, the first terms of [tex]\mathbf{A}\cdot \nabla[/tex] and [tex]\nabla\cdot \mathbf{A}[/tex] are [tex]A_1\frac{\partial}{\partial x}[/tex] and [tex]\frac{\partial}{\partial x}A_1[/tex], respectively, which are not the same.
     
  4. Dec 28, 2008 #3
    Re: Curl Product Rule confusion???

    I still don't see it, why did you add a prime symbol?

    If A is some vector with components [itex]<A_x, A_y, A_z>[/itex] and the operator [itex]\nabla =<\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}>[/itex]

    oh nevermind.... this is something I don't really need to understand.

    I think my confusion stems from the inherently weird definition of [itex]\nabla[/itex].

    (keep in mind I am an engineer :smile:)
     
  5. Dec 29, 2008 #4
    Re: Curl Product Rule confusion???

    That's a comma. Without the symbols inline, it does look like a prime.

    A and the parital derivative don't commute.

    [tex]\nabla \cdot A[/tex]
    is a scalar that will act on the vector B.

    [tex]A \cdot \nabla[/tex]
    is a derivative operator, scaled by A, that can act on the vector B.

    It's goofy notation, but it's what we have.
     
  6. Dec 29, 2008 #5

    Defennder

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  7. Dec 29, 2008 #6
    Re: Curl Product Rule confusion???

    This makes more sense now. Thanks :smile:

    I will read in the morning; after that Divergence thread, I realize that I am toast at this point :smile:

    thanks for the link
     
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