Laplace-operator

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De laplace-operator, ook wel laplaciaan genoemd, is een differentiaaloperator genoemd naar de Franse wiskundige Pierre-Simon Laplace en aangeduid door het symbool ∆. In de natuurkunde vindt de operator toepassing bij de beschrijving van voortplanting van golven (golfvergelijking), bij warmtetransport en in de elektrostatica in de laplace-vergelijking. In de kwantummechanica stelt de laplace-operator de kinetische energie voor in de schrödingervergelijking. De functies waarvoor de laplaciaan gelijk is aan nul, worden in de wiskunde harmonische functies genoemd.

Voor een scalaire functie f {\displaystyle f} $f$ op een n {\displaystyle n} $n$ -dimensionale euclidische ruimte is de laplace-operator gedefinieerd door:

Δ f = ∑ i = 1 n ∂ 2 f ∂ x i 2 {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}}

\Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}

Hierin staat ∂ 2 ∂ x i 2 {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}^{2}}}} ${\frac {\partial ^{2}}{\partial x_{i}^{2}}}$ voor de tweede partiële afgeleide naar de variabele x i {\displaystyle x_{i}} $x_{i}$ .

Als operator schrijft men daarom wel:

Δ = ∑ i = 1 n ∂ 2 ∂ x i 2 {\displaystyle \Delta =\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}}

\Delta =\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}

Alternatief kan men schrijven:

Δ f = div ⁡ ( grad ⁡ f ) {\displaystyle \Delta f=\operatorname {div} (\operatorname {grad} f)}

\Delta f=\operatorname {div} (\operatorname {grad} f)

Ook kan de laplace-operator (in rechthoekige coördinaten) uitgedrukt worden in de operator nabla (∇):

Δ = ∇ 2 = ∇ ⋅ ∇ {\displaystyle \Delta =\nabla ^{2}=\nabla \cdot \nabla }

\Delta =\nabla ^{2}=\nabla \cdot \nabla

Laplaciaan in drie dimensies

In cartesische coördinaten,

Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}

\Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}

In cilindercoördinaten:

Δ f = 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2 {\displaystyle \Delta f={1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}}

\Delta f={1 \over \rho }{\partial  \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \varphi ^{2}}+{\partial ^{2}f \over \partial z^{2}}

In bolcoördinaten:

Δ f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 {\displaystyle \Delta f={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}}

\Delta f={1 \over r^{2}}{\partial  \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial  \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}

Voorbeeld

Zij f : R 3 → R {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} } $f:\mathbb {R} ^{3}\to \mathbb {R}$ de functie gedefinieerd door

f ( x , y , z ) = x 2 + y z 2 {\displaystyle f(x,y,z)=x^{2}+yz^{2}}

f(x,y,z)=x^{2}+yz^{2}

Dan geldt:

Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 = 2 + 0 + 2 y = 2 + 2 y {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=2+0+2y=2+2y}

\Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=2+0+2y=2+2y

Laplace-operator voor een vectorveld

Voor een vectorveld A {\displaystyle A} $A$ is de laplace-operator gedefinieerd als:

Δ A = ∇ ( ∇ ⋅ A ) − ∇ × ( ∇ × A ) = grad ⁡ ( div ⁡ A ) − rot ⁡ ( rot ⁡ A ) {\displaystyle \Delta A=\nabla (\nabla \cdot A)-\nabla \times (\nabla \times A)=\operatorname {grad} (\operatorname {div} A)-\operatorname {rot} (\operatorname {rot} A)}

\Delta A=\nabla (\nabla \cdot A)-\nabla \times (\nabla \times A)=\operatorname {grad} (\operatorname {div} A)-\operatorname {rot} (\operatorname {rot} A)

In gewone cartesische coördinaten is het het vectorveld met als componenten de laplaciaan van de componenten van A , {\displaystyle A,} $A,$ dus:

Δ A = Δ = = {\displaystyle \Delta A=\Delta {\begin{bmatrix}A_{x}\\A_{y}\\A_{z}\end{bmatrix}}={\begin{bmatrix}\Delta A_{x}\\\Delta A_{y}\\\Delta A_{z}\end{bmatrix}}={\begin{bmatrix}{\frac {\partial ^{2}A_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial z^{2}}}\\{\frac {\partial ^{2}A_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial z^{2}}}\\{\frac {\partial ^{2}A_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{z}}{\partial z^{2}}}\end{bmatrix}}}

\Delta A=\Delta {\begin{bmatrix}A_{x}\\A_{y}\\A_{z}\end{bmatrix}}={\begin{bmatrix}\Delta A_{x}\\\Delta A_{y}\\\Delta A_{z}\end{bmatrix}}={\begin{bmatrix}{\frac {\partial ^{2}A_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{x}}{\partial z^{2}}}\\{\frac {\partial ^{2}A_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{y}}{\partial z^{2}}}\\{\frac {\partial ^{2}A_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}A_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}A_{z}}{\partial z^{2}}}\end{bmatrix}}

Unicode

De laplace-operator is opgenomen in Unicode als U+2206.