Einstein in 1916 developed the following notation scheme when manipulating expressions involving vectors, matrices, or tensors in general.
The rules of summation convention are:

Each index can appear at most twice in any term.

Repeated indices are implicitly summed over.

Each term must contain identical nonrepeated indices.
For example:
is a valid expression but
is not a valid expression in summation convention, since the index j appears three times in the first term.
Einstein summation implies the following;
For any vector quantity, V,
And the dot product of two vector quantity V and F is,
In a Cartesian basis, the gradient of a scalar (φ) and the divergence of a vector D can be variously written as,
We can also easily get the matrix multiplication using the Einstein summation convection;
If an N × N matrix C is the product of an N × M matrix A and an M × N matrix B
The Kronecker delta symbol δij is defined as,
And LeviCivita` permutation symbol ε_{ijk} is,
Notice that ‘1’ is for cyclic permutation of x, y, and z and ‘1’ is for anticyclic permutation of x, y, and z. Zero is when, either i=j, j=k, or k=i.
That is, if we define x=1, y=2 and z=3 then Levi Civita symbol holds following,
All other components are zero.
Using ε_{ijk} we can write index expressions for the cross product and curl. The ith component of the cross product is given by,
Now, [U✕V]_{1} = 𝜺_{1jk}U_{j}V_{k} = 𝜺_{123}U_{2}V_{3 }+ 𝜺_{132}U_{3}V_{2} + all other terms zero
This gives [U✕V]_{1} = U_{2}V_{3}  U_{3}V_{2}
Similarly,
[U✕V]_{2} = U_{3}V_{1}  U_{1}V_{3}
And [U✕V]_{3} = U_{1}V_{2}  U_{2}V_{1}
Some other standard relations defined are,
𝝏_{k}r_{j} means 𝝏_{k}/r_{j}
Additionally, 𝜺_{ijk}𝜺_{ist} = 𝜹_{js}𝜹_{kt}  𝜹_{jt}𝜹_{ks}
𝜺_{ijk} =  𝜺_{jik}
▽ = 𝝏_{i} and for any vector A, ▽A = 𝝏_{i}A_{i}.