Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:
where we remember that the raised indices refer dual vector.

$T_{i_{1}…i_{m}}=T(e_{i_{1}},…,e_{i_{m}},e_{j_{1}},…,e_{j_{m}})$

A linear map $A$ can be seen as a (1,1) tensor because:
is a number, $v∗$. is a dual vector, and W is a vector. Furthermoe, $T$ is linear in both $v∗$ and $w$. All of this makes $T$ fullfill the definition of a (1,1) tensor.

$T(w,v∗)=v∗Aw$

Bibliography:

$T_{(m,n)}$ has order $(m,n)$

The Wikipedia page of this article is basically a masterclass why Wikipedia is useless for learning technical subjects. They are not even able to teach such a simple subject properly there!

Bibliography:

- www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf gives a definition that does not consider upper and lower indexes, it only counts how many times the indices appearTheir definition of the Laplacian is a bit wrong as only one $i$ appears in it, they likely meant to have written $∂x_{i}∂ ∂x_{i}∂F $ instead of $∂x_{i}∂_{2}F $, related:

TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf.

Vectors with the index on top such as $x_{i}$ are the "regular vectors", they are called covariant vectors.

Those in indices on bottom are called contravariant vectors.

It is possible to change between them by Raising and lowering indices.

The values are different only when the metric signature matrix is different from the identity matrix.

Then a specific metric is involved, sometimes we want to automatically add it to products.

E.g., in a context considering the common Minkowski inner product matrix where the $η$ 4x4 matrix and $μ$ is a vector in $R_{4}$
which leads to the change of sign of some terms.

$x_{μ}x_{μ}=x_{μ}η_{μν}x_{ν}=−x_{0}+x_{1}+x_{2}+x_{3};$

The Einstein summation convention works will with partial derivatives and it is widely used in particle physics.

In particular, the divergence and the Laplacian can be succintly expressed in this notation:

In order to expresse partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variales with indices such as $x_{0}$, $x_{1}$, $x_{2}$, $∂_{0}$, $∂_{1}$ and $∂_{2}$ instead of the usual letters $x$, $y$ and $z$.

First we write a vector field as:
Note how we are denoting each component of $F$ as $F_{i}$ with a raised index.

$F(x_{0},x_{1},x_{2})=(F_{0}(x_{0},x_{1},x_{2}),F_{1}(x_{0},x_{1},x_{2}),F_{2}(x_{0},x_{1},x_{2})):R_{3}→R_{3}$

Then, the divergence can be written in Einstein notation as:

$∇⋅F=∂x_{0}∂F_{0}(x_{0},x_{1},x_{2}) +∂x_{1}∂F_{1}(x_{0},x_{1},x_{2}) +∂x_{2}∂F_{2}(x_{0},x_{1},x_{2}) =∂_{i}F_{i}(x_{0},x_{1},x_{2})=∂x_{i}∂F_{i}(x_{0},x_{1},x_{2}) $

It is common to just omit the variables of the function, so we tend to just say:
or equivalently when referring just to the operator:

$∇⋅F=∂_{i}F_{i}$

$∇⋅=∂_{i}$

Consider a real valued function of three variables:

$F(x_{0},x_{1},x_{2})=:R_{3}→R$

Its Laplacian can be written as:

$∇_{2}F(x_{0},x_{1},x_{2})=∂_{0}F(x_{0},x_{1},x_{2})+∂_{1}F(x_{0},x_{1},x_{2})+∂_{2}F(x_{0},x_{1},x_{2})=∂_{0}∂_{0}F(x_{0},x_{1},x_{2})+∂_{1}∂_{1}F(x_{0},x_{1},x_{2})+∂_{2}∂_{2}F(x_{0},x_{1},x_{2})=∂_{i}∂_{i}F(x_{0},x_{1},x_{2})$

It is common to just omit the variables of the function, so we tend to just say:
or equivalently when referring just to the operator:

$∇_{2}F=∂_{i}∂_{i}F$

$∇_{2}=∂_{i}∂_{i}$

Given the function $ψ$:
the operator can be written in Planck units as:
often written without function arguments as:
Note how this looks just like the Laplacian in Einstein notation, since the D'alembert operator is just a generalization of the laplace operator to Minkowski space.

$ψ:R_{4}→C$

$∂_{i}∂_{i}ψ(x_{0},x_{1},x_{2},x_{3})−m_{2}ψ(x_{0},x_{1},x_{2},x_{3})=0$

$∂_{i}∂_{i}ψ$

The Klein-Gordon equation can be written in terms of the D'alembert operator as:
so we can expand the D'alembert operator in Einstein notation to:

$□ψ+m_{2}ψ=0$

$∂_{i}∂_{i}ψ−m_{2}ψ=0$