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Coordinate transformation of contravariant vectors.

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Apr19-09, 09:53 PM
P: 77
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?
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Apr19-09, 10:06 PM
P: 896
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.
Apr20-09, 10:13 AM
P: 77
Ok thanks, that makes sense now.

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