PlanetPhysics/Laplacian

The Laplacian is a [[../Vectors/|vector]] differential [[../QuantumSpinNetworkFunctor2/|operator]]. Like all vector [[../QuantumOperatorAlgebra4/|operators]], it is given in different forms in different coordinate [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]]. In general it is given by:

${\mathrm{\nabla }}^{2}f=\mathrm{\Delta }f=\sum _i\frac{\partial f_i}{\partial x_i^2}$

where the subscript $i$ refers to the different coordinate components of the vector $f$.

Laplacian in Cartesian coordinates

As usual with vector operators, the Cartesian form is the easiest to remember and apply.

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

Laplacian in spherical coordinates

$\underset{sph}{\overset{2}{\mathrm{\nabla }}}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial }{\partial \theta }\right)+\frac{1}{r^{2}si{n}^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}$

Laplacian in cylindrical coordinates

${\mathrm{\nabla }}^2=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\theta }^2}+\frac{{\partial }^2}{\partial z^2}$