Laplacian

The Laplacian is an operator in vector calculus. It is denoted using a nabla, $\nabla ^{2}$ For some scalar function that takes in $n$ variables, $f(x_{1},x_{2},x_{3},...x_{n})$, the Laplacian may be defined as:

$\nabla ^{2}f=\sum \limits _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}={\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+...+{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}$

In Cartesian coordinates, the function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z)} . The Laplacian is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} }

See also

• Curl
• Divergence
• Gradient