Trečiadienis , 19 vasario 2020

Laplaso operatorius

Laplaso operatorius arba laplasianas – diferencialinis operatorius, atvaizduojantis skaliarinį lauką į kitą skaliarinį lauką – pirmojo lauko gradiento divergenciją.

Laplaso operatoriaus apibendrinimas vektoriniams laukams yra vektorinis Laplaso operatorius.

Laplaso operatorius pavadintas pagerbiant Pierre-Simon Laplace (1749–1827).

Skaliarinio lauko  { {\displaystyle f=\left(x_{1},x_{2},\ldots ,x_{n}\right)} }  laplasianas žymimas  { {\displaystyle \Delta f} }  arba  { {\displaystyle \nabla ^{2}f} }:

{ {\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f=\operatorname {div} (\operatorname {grad} f)=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}.} }

Fizikoje Laplaso operatorius naudojamas aprašyti bangos sklidimą, difuziją ir pan.

Laplaso operatorius stačiakampėje koordinačių sistemoje:

{ {\displaystyle \operatorname {div} \;\operatorname {grad} f=\nabla ^{2}f=\Delta f={\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}.} }

Laplaso operatorius cilindrinėje koordinačių sistemoje:

{ {\displaystyle \Delta ={1 \over \rho }{\partial \over \partial \rho }(\rho {\partial \over \partial \rho })+{1 \over \rho ^{2}}{\partial ^{2} \over \partial \phi ^{2}}+{\partial ^{2} \over \partial z^{2}}={1 \over \rho }({\partial \rho \over \partial \rho }{\partial \over \partial \rho }+\rho {\partial ^{2} \over \partial \rho ^{2}})+{1 \over \rho ^{2}}{\partial ^{2} \over \partial \phi ^{2}}+{\partial ^{2} \over \partial z^{2}},} }

{ {\displaystyle \Delta f={1 \over \rho }{\partial f \over \partial \rho }+{\partial ^{2}f \over \partial \rho ^{2}}+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}.} }

Laplaso operatorius sferinėje koordinačių sistemoje:

{ {\displaystyle \Delta ={1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial \over \partial r})+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }(\sin \theta {\partial \over \partial \theta })+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2} \over \partial \phi ^{2}},} }

{ {\displaystyle \Delta f={1 \over r^{2}}(2r{\partial f \over \partial r}+r^{2}{\partial ^{2}f \over \partial r^{2}})+{1 \over r^{2}\sin \theta }(\cos \theta {\partial f \over \partial \theta }+\sin \theta {\partial ^{2}f \over \partial \theta ^{2}})+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}=} }

{ {\displaystyle ={2 \over r}{\partial f \over \partial r}+{\partial ^{2}f \over \partial r^{2}}+{\cot \theta \over r^{2}}{\partial f \over \partial \theta }+{1 \over r^{2}}{\partial ^{2}f \over \partial \theta ^{2}}+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}.} }

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