Suppose I have a vector in the covariant basis $\bar e_i$ and a metric $g$ and I want to obtain a unit vector $\bar u_i$ parallel to $\bar e_i$. I would write:
$\bar e_i = \hat u_i ||\bar e_i|| = \hat u_i \sqrt {\bar e_i \bullet \bar e_i} = \hat u_i \sqrt{g_{ii}} $
This seems weird because the index $i$ appears a bunch of times but I guess it's okay because it's a free index?
Now suppose I want to calculate the gradient in terms of the vectors $\hat u_i$. We have: $\mathbf {\nabla} = \vec{e}^j\frac{\partial}{\partial x^j} = g^{ji} \bar{e}_i \frac{\partial}{\partial x^j} = g^{ji}\sqrt{g_{ii}}\hat u_i\frac{\partial}{\partial x^j} $
Now that's definitely wrong. The dummy index $j$ appears twice, up and down so this is okay, but the dummy index $i$ appears four times! This breaks the rules. I could maybe rewrite it as:
$\mathbf {\nabla} = \vec{e}^j\frac{\partial}{\partial x^j} = g^{ji} \bar{e}_i \frac{\partial}{\partial x^j} = g^{ji}\sqrt{g_{ik}}\hat u_k\frac{\partial}{\partial x^j} $
This now satisfies the rules, but it isn't the same equation. I really want the diagonal element $g_{ii}$, not $g_{ik}$. What am I supposed to do?