: Mathematics: Laplacian

mathematical physics > differential operator > Laplacian

mathematical analysis > functional analysis > operator > differential operator > Laplacian

mathematical physics > vector calculus > vector calculus identities > Laplacian

mathematical analysis > calculus > vector calculus > vector calculus identities > Laplacian

geometry > differential geometry > vector calculus > vector calculus identities > Laplacian

Laplacian

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ${\\displaystyle \ abla \\cdot \ abla }$ , ${\\displaystyle \ abla ^{2}}$ (where ${\\displaystyle \ abla }$ is the nabla operator), or ${\\displaystyle \\Delta }$ . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian $Δ f (p)$ of a function $f$ at a point $p$ measures by how much the average value of $f$ over small spheres or balls centered at $p$ deviates from $f (p)$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Laplace_operator)

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