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Ridiculous Question (Derivatives)

  1. Aug 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Under what conditions may I change temporal and spatial derivatives? I cannot remember for the life of me.

    EG:
    [tex]\frac{\partial }{\partial t}\left( \nabla \cdot \mathbf{A} \right) = \nabla \cdot \left( \frac{\partial \mathbf{A}}{\partial t}\right)[/tex]

    Thanks.
     
  2. jcsd
  3. Aug 14, 2009 #2
    partial derivative? Any time you want, I think.
     
  4. Aug 14, 2009 #3
    That's what I thought, too, but I was unsure. Trying to de-rust, here, heh.
     
  5. Aug 14, 2009 #4
    Go back to the definition of the divergence, so that your equation contains terms like

    [tex] \frac {\partial } {\partial t} \frac {\partial } {\partial x} \mathbf{A}_x \right) \ . [/tex]

    While x and t are independent variables,

    [tex] \frac {\partial } {\partial t} \frac {\partial } {\partial x} = \frac {\partial } {\partial x} \frac {\partial } {\partial t} \ .[/tex]
     
  6. Aug 14, 2009 #5

    Redbelly98

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    This works for gradient and curl too, by the way. It's basically a matter of


    ∂²A/∂x∂t = ∂²A/∂t∂x

    and similarly for y and z.
     
  7. Aug 14, 2009 #6
    Could you use this to show that electric and magnetic fields don't move in space?
     
  8. Aug 14, 2009 #7

    Redbelly98

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    Hmmm, not sure I understand your question ... E-M fields can and do move through space.
     
  9. Aug 14, 2009 #8
    The value of the fields change over time for any given coordinate, but the fields are not considered moving objects. The field is attached to a coordinate. A propagating electromagnetic wave is like a sound wave. The wave moves, the molecules stay (nominally) in place. I may be confused about the math, though.
     
  10. Aug 14, 2009 #9

    Redbelly98

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    Okay. Well, if you're going to say by definition that the fields don't move through space, not sure why you'd need a proof of that.

    Or even if you're not saying this is by definition, I don't see how my earlier statement in post #5 could be used to prove it. That statement is a basic consequence of multivariable calculus.
     
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