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Coordinate transformation of contravariant vectors.

  1. Apr 19, 2009 #1

    trv

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    Note: The derivatives are partial.

    I've seen the coordinate transformation equation for contravariant vectors given as follows,

    V'a=(dX'a/dXb)Vb

    What I don't get is the need for two indices a and b. Wouldn't it be adequate to just write the equation as follows?

    V'a=(dX'a/dXa)Va

    The prime being adequate to indicate the new and the unprimed the old, coordinates and contravariant vector. Or does the second index provide some more information which I am unaware of?
     
  2. jcsd
  3. Apr 19, 2009 #2
    The first equation has on the LHS a single component of V' while the RHS is a sum by summation convention over all the unprimed components.
    [tex]V'^1 = \frac{\partial X'^1}{\partial X^1}V^1 + \cdots + \frac{\partial X'^1}{\partial X^n}V^n\\
    \vdots
    V'^m = \frac{\partial X'^m}{\partial X^1}V^1 + \cdots + \frac{\partial X'^m}{\partial X^n}V^n
    [/tex]
    Your equation is a single component and represents no sum, so it is not equivalent.
    [tex]V'^1 = \frac{\partial X'^1}{\partial X^1}V^1
    \vdots
    V'^m = \frac{\partial X'^m}{\partial X^m}V^m[/tex]
    It seems to state that the ath component of V' depends only on the ath component of V, which is usually not the case.
     
    Last edited: Apr 19, 2009
  4. Apr 20, 2009 #3

    trv

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    Ok thanks, that makes sense now.
     
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