OurBigBook.com $£ Sponsor 中国独裁统治 China Dictatorship 新疆改造中心、六四事件、法轮功、郝海东、709大抓捕、2015巴拿马文件邓家贵、低端人口、西藏骚乱

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

 Ancestors (8)

 Incoming links (2)