# Coordinate transformation of contravariant vectors.

by trv
Tags: contravariant, coordinate, transformation, vectors
 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?
 P: 855 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. $$V'^1 = \frac{\partial X'^1}{\partial X^1}V^1 + \cdots + \frac{\partial X'^1}{\partial X^n}V^n\\ \vdots V'^m = \frac{\partial X'^m}{\partial X^1}V^1 + \cdots + \frac{\partial X'^m}{\partial X^n}V^n$$ Your equation is a single component and represents no sum, so it is not equivalent. $$V'^1 = \frac{\partial X'^1}{\partial X^1}V^1 \vdots V'^m = \frac{\partial X'^m}{\partial X^m}V^m$$ It seems to state that the ath component of V' depends only on the ath component of V, which is usually not the case.
 P: 77 Ok thanks, that makes sense now.

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