Laplaceov operator

Laplaceov operátor [laplásov ~] (tudi laplacian in redkeje operator delta) je v vektorskem računu skalarni diferencialni operator skalarne funkcije φ. Je enak vsoti vseh drugih parcialnih odvodov odvisne spremenljivke.

To odgovarja div (grad φ), zato tudi uporaba simbola del (operator nabla), ki ga predstavlja:

$\nabla ^{2}\phi =\nabla \cdot (\nabla \phi )\!\,.$

Zapišemo ga tudi z znakom Δ.

V eno in dvorazsežnih kartezičnih koordinatah je Laplaceov operator:

$\Delta _{1}\equiv \nabla _{1}^{2}={\partial ^{2} \over \partial x^{2}}\;,\quad \Delta _{2}\equiv \nabla _{2}^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}\;.$

In v treh Σ(x, y, z):

$\Delta _{3}\equiv \nabla _{3}^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}\;.$

V trorazsežnih cilindričnih koordinatah Σ(r, φ, z) je:

$\nabla ^{2}t={1 \over r}{\partial \over \partial r}\left(r{\partial t \over \partial r}\right)+{1 \over r^{2}}{\partial ^{2}t \over \partial \phi ^{2}}+{\partial ^{2}t \over \partial z^{2}}$

V trorazsežnih sferičnih koordinatah Σ(r, θ, φ) je:

$\nabla ^{2}t={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial t \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial t \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}t \over \partial \phi ^{2}}$

Laplaceov operator se na primer pojavlja v Laplaceovi, Poissonovi, Poisson-Boltzmannovi, Helmholtzovi ali valovni enačbi.

Laplaceov operator je linearen:

$\nabla ^{2}(f+g)=\nabla ^{2}f+\nabla ^{2}g\!\,.$

Velja tudi:

$\nabla ^{2}(fg)=(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)\!\,.$