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In vector calculus, the Laplace operator or
Laplacian is a differential operator
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:
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It is also written as Δ.
In two-dimensional Cartesian coordinates, the
Laplacian is:
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In three-dimensional Cartesian coordinates:
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In cylindrical coordinates:
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In spherical coordinates:
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The Laplacian is linear:
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The following holds also:
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It occurs in Laplace's equation and Poisson's equation.
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