Coordinate transformation of contravariant vectors.

[trv]

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?

 

[slider142]

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.
$$V'^1 = \frac{\partial X'^1}{\partial X^1}V^1 + \cdots + \frac{\partial X'^1}{\partial X^n}V^n\\<br /> \vdots<br /> V'^m = \frac{\partial X'^m}{\partial X^1}V^1 + \cdots + \frac{\partial X'^m}{\partial X^n}V^n$$
Your equation is a single component and represents no sum, so it is not equivalent.
$$V'^1 = \frac{\partial X'^1}{\partial X^1}V^1<br /> \vdots<br /> V'^m = \frac{\partial X'^m}{\partial X^m}V^m$$
It seems to state that the ath component of V' depends only on the ath component of V, which is usually not the case.

 

[trv]

Ok thanks, that makes sense now.