Solving Einstein Notation: Summing Subscripts and Superscripts

[andrewkirk]

$\vec{v} = v^i\vec{e}_i = g(\vec{v},\vec{e}_i)\vec{e}_i$

The last bit is a sum over i but will need a Ʃ because the Einstein rule only applies to matched superscripts and subscripts and here bot the i are subscripts.

Even if I write out the metric in the basis it doesn't work:

$g(\vec{v},\vec{e}_i)\vec{e}_i=g_{ab}v^ae^{b}_{i} \vec{e}_i$

In everything else I've ever done the indices have always been where they needed to be for Einstein summation but for some reason in this one they're not. It's no hardship to write the $\Sigma^{n}_{i=1}$ before it but it just feels as though there should be a way to avoid that.

Any suggestions or comments? Thanks very much.

 

[Ben Niehoff]

You can use the dual basis, defined by

$$e^i (e_j) = \delta^i{}_j$$
Then you have

$$v = v^i e_i = e^i (v) \, e_i$$

 

[andrewkirk]

Thanks very much. I thought there had to be a way.

$\vec{v}=v^i\vec{e}_i=\tilde{e}^i(\vec{v})\vec{e}_i$

Beautiful!