Laplaceov operator

Laplasov operator, u matematici, je eliptički diferencijalni operator drugog reda. Ima brojne primene širom matematike, te u fizici, elektrostatici, kvantnoj mehanici, obradi snimaka, itd. Nazvan je po francuskom matematičaru Pjeru Simonu Laplasu.

Imajući u vidu pojmove divergencije i gradijenta, za datu skalarnu funkciju u = u ( x , y , z ) {\displaystyle u=u(x,y,z)} , biće:

d i v g r a d u = ( ∂ 2 u ∂ x 2 , ∂ 2 u ∂ y 2 , ∂ 2 u ∂ z 2 ) {\displaystyle div\,grad\,u=({\frac {\partial ^{2}u}{\partial x^{2}}},{\frac {\partial ^{2}u}{\partial y^{2}}},{\frac {\partial ^{2}u}{\partial z^{2}}})} ,

što se može napisati kao:

d i v g r a d u = ( ∂ 2 ∂ x 2 , ∂ 2 ∂ y 2 , ∂ 2 ∂ z 2 ) u {\displaystyle div\,grad\,u=({\frac {\partial ^{2}}{\partial x^{2}}},{\frac {\partial ^{2}}{\partial y^{2}}},{\frac {\partial ^{2}}{\partial z^{2}}})u} .

Desna strana poslednjeg izraza, bez oznake za funkciju u {\displaystyle u} , predstavlja Laplasov operator i obeležava se sa delta - Δ:

Δ = ( ∂ 2 ∂ x 2 , ∂ 2 ∂ y 2 , ∂ 2 ∂ z 2 ) {\displaystyle \Delta =({\frac {\partial ^{2}}{\partial x^{2}}},{\frac {\partial ^{2}}{\partial y^{2}}},{\frac {\partial ^{2}}{\partial z^{2}}})} .

Koristeći operator nabla, taj izraz možemo zapisati kao:

∇ 2 ϕ = ∇ ⋅ ( ∇ ϕ ) . {\displaystyle \nabla ^{2}\phi =\nabla \cdot (\nabla \phi )\;.}

Koordinatni izrazi

U jednodimenzionalnom i dvodimenzionalnom Dekartovom koordinatnom sistemu Laplasov operator je:

Δ 1 ≡ ∇ 1 2 = ∂ 2 ∂ x 2 , Δ 2 ≡ ∇ 2 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 . {\displaystyle \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}}\;.}

U trodimenzionalnom Dekartovom koordinatnom sistemu je :

Δ 3 ≡ ∇ 3 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 . {\displaystyle \Delta _{3}\equiv \nabla _{3}^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}\;.}

U trodimenzionalnom cilindričnom koordinatnom sistemu je:

∇ 2 t = 1 r ∂ ∂ r ( r ∂ t ∂ r ) + 1 r 2 ∂ 2 t ∂ ϕ 2 + ∂ 2 t ∂ z 2 {\displaystyle \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}}}

U trodimenzionalnom sfernom koordinatnom sistemu je :

∇ 2 t = 1 r 2 ∂ ∂ r ( r 2 ∂ t ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ t ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 t ∂ ϕ 2 {\displaystyle \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}}}

U Euklidskom prostoru R n {\displaystyle {\mathbb {R} }^{n}} Laplasov operator je dat u standardnim koordinatama kao

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

Laplasov operator u opštim krivolinijskim koordinatama dan je sa:

∇ 2 f ( q 1 ,   q 2 ,   q 3 ) = div grad f ( q 1 ,   q 2 ,   q 3 ) = {\displaystyle \nabla ^{2}f(q_{1},\ q_{2},\ q_{3})=\operatorname {div} \,\operatorname {grad} \,f(q_{1},\ q_{2},\ q_{3})=}
= 1 H 1 H 2 H 3 [ ∂ ∂ q 1 ( H 2 H 3 H 1 ∂ f ∂ q 1 ) + ∂ ∂ q 2 ( H 1 H 3 H 2 ∂ f ∂ q 2 ) + ∂ ∂ q 3 ( H 1 H 2 H 3 ∂ f ∂ q 3 ) ] , {\displaystyle ={\frac {1}{H_{1}H_{2}H_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {H_{2}H_{3}}{H_{1}}}{\frac {\partial f}{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {H_{1}H_{3}}{H_{2}}}{\frac {\partial f}{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {H_{1}H_{2}}{H_{3}}}{\frac {\partial f}{\partial q_{3}}}\right)\right],}
gde su H i   {\displaystyle H_{i}\ } Lameovi koeficijenti.

U slučaju Rimanovoga krivolinijskoga prostora definisanoga metričkim tenzorom g i j {\displaystyle g_{ij}} Laplasijan je dan sa:

∇ 2 f = 1 g ∑ i = 1 n ∂ ∂ x i ( g ∑ k = 1 n g i k ∂ f ∂ x k ) {\displaystyle \nabla ^{2}f={\frac {1}{\sqrt {g}}}\sum _{i=1}^{n}{\frac {\partial }{\partial x^{i}}}({\sqrt {g}}\sum _{k=1}^{n}g^{ik}{\frac {\partial f}{\partial x^{k}}})}

a metrika prostora definisana je sa:

d s 2 = ∑ i , j = 1 n g i j d x i d x j {\displaystyle ds^{2}=\sum _{i,j=1}^{n}g_{ij}dx^{i}dx^{j}} .

Svojstva

Laplasov operator je linearan:

∇ 2 ( f + g ) = ∇ 2 f + ∇ 2 g . {\displaystyle \nabla ^{2}(f+g)=\nabla ^{2}f+\nabla ^{2}g\;.}

Takođe važi :

∇ 2 ( f g ) = ( ∇ 2 f ) g + 2 ( ∇ f ) ⋅ ( ∇ g ) + f ( ∇ 2 g ) . {\displaystyle \nabla ^{2}(fg)=(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)\;.}

Uopštenja

Laplasov operator se može uopštiti na više načina. Dalamberov operator je definisan na prostoru Minkovskog. Laplas-Beltramijev operator je eliptički diferencijalni operator drugog reda definisan na svakoj Rimanovoj mnogostrukosti. Laplas-de Ramov operator dejstvuje na prostorima diferencijalnih formi na pseudo-Rimanovim površima.

Literatura

  • Evans, L (1998), Partial Differential Equations, American Mathematical Society, ISBN 978-0-8218-0772-9 .
  • Feynman, R, Leighton, R, and Sands, M (1970), „Chapter 12: Electrostatic Analogs”, The Feynman Lectures on Physics, Volume 2, Addison-Wesley-Longman .
  • Gilbarg, D.; Trudinger, N. (2001), Elliptic partial differential equations of second order, Springer, ISBN 978-3-540-41160-4 .
  • Schey, H. M. (1996), Div, grad, curl, and all that, W W Norton & Company, ISBN 978-0-393-96997-9 .

Spoljašnje veze

  • Hazewinkel Michiel, ur. (2001). „Laplace operator”. Encyclopaedia of Mathematics. Springer. ISBN 978-1-55608-010-4. 
  • Weisstein, Eric W., "Laplacian", MathWorld.