High School Einstein summation notation, ambiguity?

[Hiero]

If I see $f(x_ie_i)$ I assume it means $f(\Sigma x_ie_i)$ (summing in the domain of f) but what if I instead wanted to write $\Sigma f(x_ie_i)$ (summing in the range)?

Is there a way to distinguish between these in Einstein’s summation notation?

 

[PAllen]

This is an odd case, but I think if you want the latter, in a context where Einstein summation is implied, you have to write it explicitly, overriding the convention. Only first case is handled by the summation convention.

 

[Dale]

Is there a way to distinguish between these in Einstein’s summation notation?

So $f(\Sigma x^{\mu}e_{\mu})$ is $f(x^{\mu}e_{\mu})$ and $\Sigma f(x^{\mu}e_{\mu})$ is $f(x^{\mu}e_{\nu})\delta^{\nu}_{\mu}$

 

[Hiero]

So $f(\Sigma x^{\mu}e_{\mu})$ is $f(x^{\mu}e_{\mu})$ and $\Sigma f(x^{\mu}e_{\mu})$ is $f(x^{\mu}e_{\nu})\delta^{\nu}_{\mu}$

Oh wow, that’s clever! Thanks for the insight.