Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Coordinate transformation of contravariant vectors.

  1. Apr 19, 2009 #1


    User Avatar

    Note: The derivatives are partial.

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


    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?


    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\\
    V'^m = \frac{\partial X'^m}{\partial X^1}V^1 + \cdots + \frac{\partial X'^m}{\partial X^n}V^n
    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
    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


    User Avatar

    Ok thanks, that makes sense now.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook