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Laplacian in Einstein notation ( $\partial_{i} \partial^{i}$ )

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:

\nabla^{2} 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:

\nabla^{2} F = \partial_{i} \partial^{i} F

or equivalently when referring just to the operator:

\nabla^{2} = \partial_{i} \partial^{i}

Table of contents 188 2
- D'alembert operator in Einstein notation ( $\partial_{i} \partial^{i}$ ) Laplacian in Einstein notation 88 1
  - Klein-Gordon equation in Einstein notation D'alembert operator in Einstein notation 30

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