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 Pjerą Simoną Laplasą (1749–1827). Taikomas diferencialinių lygčių teorijoje, funkcijų teorijoje, funkcinėje analizėje ir geometrijoje. Skaliarinio lauko f = ( x 1 , x 2 , … , x n ) {\displaystyle f=\left(x_{1},x_{2},\ldots ,x_{n}\right)} laplasianas žymimas Δ f {\displaystyle \Delta f} arba ∇ 2 f {\displaystyle \nabla ^{2}f} : Δ f = ∇ 2 f = ∇ ⋅ ∇ f = div ⁡ ( grad ⁡ f ) = ∑ i = 1 n ∂ 2 f ∂ x i 2 . {\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: div grad ⁡ f = ∇ 2 f = Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\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: Δ = 1 ρ ∂ ∂ ρ ( ρ ∂ ∂ ρ ) + 1 ρ 2 ∂ 2 ∂ ϕ 2 + ∂ 2 ∂ z 2 = 1 ρ ( ∂ ρ ∂ ρ ∂ ∂ ρ + ρ ∂ 2 ∂ ρ 2 ) + 1 ρ 2 ∂ 2 ∂ ϕ 2 + ∂ 2 ∂ z 2 , {\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}},} Δ f = 1 ρ ∂ f ∂ ρ + ∂ 2 f ∂ ρ 2 + 1 ρ 2 ∂ 2 f ∂ ϕ 2 + ∂ 2 f ∂ 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: Δ = 1 r 2 ∂ ∂ r ( r 2 ∂ ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 ∂ ϕ 2 , {\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}},} Δ f = 1 r 2 ( 2 r ∂ f ∂ r + r 2 ∂ 2 f ∂ r 2 ) + 1 r 2 sin ⁡ θ ( cos ⁡ θ ∂ f ∂ θ + sin ⁡ θ ∂ 2 f ∂ θ 2 ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ ϕ 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}}=} = 2 r ∂ f ∂ r + ∂ 2 f ∂ r 2 + cot ⁡ θ r 2 ∂ f ∂ θ + 1 r 2 ∂ 2 f ∂ θ 2 + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ ϕ 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}}.}