Einstein notation for partial derivatives

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 $x_0$, $x_1$, $x_2$, ${\partial }_0$, ${\partial }_1$ and ${\partial }_2$ 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.

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

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:

Consider a real valued function of three variables:

Its Laplacian can be written as:

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:

Given the function $\psi$: 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.

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: