Einsteins Summmation Convention

Einstein’s summation convention allows to omit sum signs. Essentially if a dummy index \(i\) occurs exactly twice in a upper and a lower index it should be summed over all possible \(i\), e.g.

\begin{equation*} x^ie_i:= \sum_{i=1}^{n} x^ie_i. \end{equation*}

The convention is, that indices are written up, if the tensor type goes up by contraction and vice versa. For example

\begin{equation*} g_{ij}x^j=\alpha_i \end{equation*}

are the components of a \(1\)-form and

\begin{equation*} g^{ij}\alpha_i=x^j \end{equation*}

are the components of a vector.