= Laplacian in Einstein notation
{c}
{title2=$\partial_i \partial^i$}
Consider a real valued function of three variables:
$$
F(x_0, x_1, x_2) = : \R^3 \to \R
$$
Its <Laplacian> can be written as:
$$
\laplacian{F(x_0, x_1, x_2)} = \\
\partial_0^2 F(x_0, x_1, x_2) + \partial_1^2 F(x_0, x_1, x_2) + \partial_2^2 F(x_0, x_1, x_2) = \\
\partial_0 \partial^0 F(x_0, x_1, x_2) + \partial_1 \partial^1 F(x_0, x_1, x_2) + \partial_2 \partial^2 F(x_0, x_1, x_2) = \\
\partial_i \partial^i F(x_0, x_1, x_2)
$$
It is common to just omit the variables of the function, so we tend to just say:
$$
\laplacian{F} = \partial_i \partial^i F
$$
or equivalently when referring just to the <operator>:
$$
\laplacian{} = \partial_i \partial^i
$$