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Coordinate transformation of contravariant vectors. 
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#1
Apr1909, 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}/dX^{b})V^{b} 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}/dX^{a})V^{a} 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
Apr1909, 10:06 PM

P: 905

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. 


#3
Apr2009, 10:13 AM

P: 77

Ok thanks, that makes sense now.



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