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 ${\displaystyle u=u(x,y,z)}$, biće:

${\displaystyle div\,grad\,u=({\frac {\partial ^{2}u}{\partial x^{2}}},{\frac {\partial ^{2}u}{\partial y^{2}}},{\frac {\partial ^{2}u}{\partial z^{2}}})}$,

što se može napisati kao:

${\displaystyle div\,grad\,u=({\frac {\partial ^{2}}{\partial x^{2}}},{\frac {\partial ^{2}}{\partial y^{2}}},{\frac {\partial ^{2}}{\partial z^{2}}})u}$.

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

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

Koristeći operator nabla, taj izraz možemo zapisati kao:

${\displaystyle \nabla ^{2}\phi =\nabla \cdot (\nabla \phi )\;.}$

Koordinatni izrazi

U jednodimenzionalnom i dvodimenzionalnom Dekartovom koordinatnom sistemu Laplasov operator je:

${\displaystyle \Delta _{1}\equiv \nabla _{1}^{2}={\partial ^{2} \over \partial x^{2}}\;,\quad \Delta _{2}\equiv \nabla _{2}^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}\;.}$

U trodimenzionalnom Dekartovom koordinatnom sistemu je :

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

U trodimenzionalnom cilindričnom koordinatnom sistemu je:

${\displaystyle \nabla ^{2}t={1 \over r}{\partial \over \partial r}\left(r{\partial t \over \partial r}\right)+{1 \over r^{2}}{\partial ^{2}t \over \partial \phi ^{2}}+{\partial ^{2}t \over \partial z^{2}}}$

U trodimenzionalnom sfernom koordinatnom sistemu je :

${\displaystyle \nabla ^{2}t={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial t \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial t \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}t \over \partial \phi ^{2}}}$

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

${\displaystyle \Delta _{n}=\nabla _{n}^{2}=\sum _{j=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}}$.

Laplasov operator u opštim krivolinijskim koordinatama dan je sa:

${\displaystyle \nabla ^{2}f(q_{1},\ q_{2},\ q_{3})=\operatorname {div} \,\operatorname {grad} \,f(q_{1},\ q_{2},\ q_{3})=}$

${\displaystyle ={\frac {1}{H_{1}H_{2}H_{3}}}\left[{\frac {\partial }{\partial q_{1}}}\left({\frac {H_{2}H_{3}}{H_{1}}}{\frac {\partial f}{\partial q_{1}}}\right)+{\frac {\partial }{\partial q_{2}}}\left({\frac {H_{1}H_{3}}{H_{2}}}{\frac {\partial f}{\partial q_{2}}}\right)+{\frac {\partial }{\partial q_{3}}}\left({\frac {H_{1}H_{2}}{H_{3}}}{\frac {\partial f}{\partial q_{3}}}\right)\right],}$

gde su ${\displaystyle H_{i}\ }$ Lameovi koeficijenti.

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

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

a metrika prostora definisana je sa:

${\displaystyle ds^{2}=\sum _{i,j=1}^{n}g_{ij}dx^{i}dx^{j}}$ .

Svojstva

Laplasov operator je linearan:

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

Takođe važi :

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

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.

Vanjske veze

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