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 $f(x,y,z)$. The Laplacian is:

$\nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}$

See also

• Curl
• Divergence
• Gradient