Einstein Summation Convention

Einstein summation convention is a mathematical tool for simplifying expressions (bigger to smaller) including summations of vectors, matrices, and general tensors.

Here, we use indices to define a particular term.

i.e.

for example: $a_{i}$ (or $a^{i}$) , where “i” is called index.

As lots of mathematical expressions involve summation over particular indices, that’s why these conventions are used widely to write bigger expressions.

Rule #1

Any twice-repeated index in a “single term” is summed over the positive integers (i=1,2,3…n), usually in n=3

Explaination:

If we have the following type of long & boring expression,

$\sum_{i=1}^{3}a_{i}x_{i}= a_{1}x_{1}+ a_{2}x_{2}+ a_{3}x_{3}$

then, using einstein summation convention, we can simply write:

$\sum_{i=1}^{3}a_{i}x_{i} = a_{i}x_{i}$ (where, $a_{i}x_{i}$ is called a “term”)

Rule #2

Each index can appear maximum 2 times in a term.

Explanation:

Example:

$a_{ij}b_{ij}$ is valid

but, $a_{ii}b_{ij}$ is invalid becuase i is appearing 3 times.