Einstein Notation

#Math #Physics

$\displaystyle a_{i}b_{i}=\sum_{i}a_{i}b_{i}$

Rules:

(1) Einstein Summation Convention
$a_ib_i=\displaystyle\sum_iab$
When there are two common indices in an expression, they may be assumed to be summed over that index
E.g.:
$\Gamma_{ijk}x_ab_c=\displaystyle\sum_{a}\displaystyle\sum_b\Gamma_{ijk}x_ab_c$
(2) Dummy Indices
For the expressions $a_ib_i + c_jd_j + e_nf_m$, $i$ 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 $a_kb_k + c_ld_l + e_nf_m$ 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
$a_ib_{ij}$ is ok
$a_ib_{ii}$ is not ok
$a^{ij}*i$ 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