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How to show something transforms as a covector?

  1. Apr 15, 2006 #1
    Considering a boost in the x direction, how do you show that


    transforms as a covector?

  2. jcsd
  3. Apr 15, 2006 #2

    Meir Achuz

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    I assume you mean partial derivs.
    Use the calculus rule relating \partial_x' to \partial_x.
  4. Apr 15, 2006 #3
    I don't see that this is a covariant vector. If it was the -gradient then I'd see that (you used total derivatives rather than partial derivative. I've never seen such an object). If it was then I'd use the chain rule for partial differentiation.

  5. Apr 15, 2006 #4
    yes sorry they are partials... why is the chain rule used.. I don't see how that leads to the solution

  6. Apr 15, 2006 #5
    [tex]\frac{\partial}{\partial x'} = \frac{\partial x}{\partial x'}\frac{\partial}{\partial x} + \frac{\partial y}{\partial y'}\frac{\partial}{\partial y} + \frac{\partial z}{\partial z'}\frac{\partial}{\partial z} + \frac{\partial t}{\partial t'}\frac{\partial}{\partial t}[/tex]

    Now change coordinates (x, y, z, t) - > (x1, x2, x3, x4)

    Then the above sum can be written as a sum which is identical to the law of covariant transformation.

    Last edited: Apr 15, 2006
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