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Coordinate transformation of contravariant vectors.

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trv
#1
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,

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?
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slider142
#2
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\\
\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.
trv
#3
Apr20-09, 10:13 AM
P: 77
Ok thanks, that makes sense now.


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