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Understanding Differential Notation

  1. Dec 20, 2005 #1
    The four velocity component [tex]u^\mu[/tex] with coordinate of [tex]x^\mu(\lambda)[/tex] is given by

    [tex]u^\mu = \frac{dx^\mu}{d\lambda}[/tex]

    where [tex]\lambda[/tex] is the proper time. So, the component of acceleration [tex]a^\mu[/tex] is

    [tex]a^\mu = \frac{du^\mu}{d\lambda}[/tex]

    Using the chain rule we have

    [tex]a^\mu = \frac{\partial u^\mu}{\partial x^\alpha} \frac{dx^\alpha}{d\lambda} = u^\alpha \partial_{\alpha}u^\mu[/tex]

    Everything was straight forward except the last part, I don't understand what the notation of [tex]\partial_{\alpha}[/tex] meant.
     
    Last edited: Dec 20, 2005
  2. jcsd
  3. Dec 20, 2005 #2

    Tom Mattson

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    [itex]\partial_{\alpha}[/itex] is the covariant 4-gradient. It means nothing other than the following:

    [tex]\partial_{\alpha}=\frac{\partial}{\partial x^{\alpha}}[/tex]
     
    Last edited: Dec 20, 2005
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