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# Laplace operator

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In vector calculus, the Laplace operator or Laplacian is a differential operator[?]. It is equal to the sum of all the second partial derivatives of a dependent variable.

This corresponds to div(grad φ), hence the use of the symbol del to represent it:

$\nabla^2 \phi = \nabla \cdot ( \nabla \phi )$

It is also written as Δ.

In two dimensional Cartesian coordinates, the Laplacian is:

$\nabla^2 = {\partial^2 \over \partial x^2 } + {\partial^2 \over \partial y^2 }$

In three:

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

It occurs, for example, in Laplace's equation.

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