Index raising and lowering: GR

[Ai52487963]

Homework Statement

Consider a 2-dimensional line element: $$ds^2=x^2dx+2dxdy-dy^2$$

Raise and lower the indices of given vectors by finding the raised index metric, etc, blah blah

Homework Equations

$$V_a = (1,-1)$$ and $$W^a = (0,1)$$

The Attempt at a Solution

Solution is given and I understand how to raise and lower indices, but I'm just wondering if my book has a massive exemplary typo. They raise V by multiplying it by the raised metric, which is fine. I get that. I don't get why they use the raised metric for lowering the W vector. Shouldn't you just use the lower metric given by the line element?

 

[clamtrox]

That's right. The whole point of the index notation is you can immediately see what you have to do.
$V^a = g^{ab} V_b, W_a = g_{ab}W^b$

Which book is that? You can usually find an errata online.

 

[Ai52487963]

That's right. The whole point of the index notation is you can immediately see what you have to do.
$V^a = g^{ab} V_b, W_a = g_{ab}W^b$

Which book is that? You can usually find an errata online.

It's Relativity Demystified by McMahon