Einstein Notation

$a_i{b}_i=\sum _i{a}_i{b}_i$

• Sums are assumed when indices are shared
• Results in much shorter equations when combined with Levi-Civita Symbol

Rules:
- (1) Einstein Summation Convention
$a_i{b}_i=\sum _{i}ab$
When there are two common indices in an expression, they may be assumed to be summed over that index
E.g.:
${\Gamma }_{ijk}{x}_a{b}_c=\sum _a\sum _b{\Gamma }_{ijk}{x}_a{b}_c$
- (2) Dummy Indices
For the expressions $a_i{b}_i+c_j{d}_j+e_n{f}_m$, $i$ and $j$ are repeated twice in a single term, so they are called dummy indices.
Dummy indicies can be interchanged for other variables, so the above expression could also be represented as $a_k{b}_k+c_l{d}_l+e_n{f}_m$ provided $i$ and $k$ have the same range and so too do $j$ and $l$
- (3) Free Indices
An index that occurs once in a term
Free indices can’t be interchanged like dummy indices can be
- (4) Repetition of Indices
Indices cannot be repeated more than twice in a term
$a_i{b}_{ij}$ is ok
$a_i{b}_{ii}$ is not ok
$a_i^{ij}$ would indicate one pair of dummy variables $i$ and one free variable $j$
- (5) Equations
Free indices in equations must match
E.g.: $x_i=a_{ij}{b}_j$
Here, $i$ is the only free variable here, and it matches on both sides
Non e.g.: $x_i=a_{ij}$
Here, $i$ and $j$ are free variables, but $j$ doesn’t match both sides