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
, biće:
,
što se može napisati kao:
.
Desna strana poslednjeg izraza, bez oznake za funkciju
, predstavlja Laplasov operator i obeležava se sa delta - Δ:
.
Koristeći operator nabla, taj izraz možemo zapisati kao:
![{\displaystyle \nabla ^{2}\phi =\nabla \cdot (\nabla \phi )\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17e150af3bb41e381a229415360d09abc474d5b)
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}}\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff91b6f94bcae4989718ef42c56bf0476787456)
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}}\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/793a38e385bf3f7b44370f326e208e05c37232d4)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e970711407510086a2e9077f6bca302c92240a1)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/576afc8a05dddf1a7f0e47c685d46cc05aaa3707)
U Euklidskom prostoru
Laplasov operator je dat u standardnim koordinatama kao
.
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})=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/916a27a604002ea6dd2ed735c17a5cedd001b700)
![{\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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20976c3be6b574fb20939f284033ac2c05f005ac)
- gde su
Lameovi koeficijenti.
U slučaju Rimanovoga krivolinijskoga prostora definisanoga metričkim tenzorom
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}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b583db3feb34988ff30cca15b47443ff4da412)
a metrika prostora definisana je sa:
.
Svojstva
Laplasov operator je linearan:
![{\displaystyle \nabla ^{2}(f+g)=\nabla ^{2}f+\nabla ^{2}g\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c01ce0d6ed32af56592d530b384592d4aac32aaf)
Takođe važi :
![{\displaystyle \nabla ^{2}(fg)=(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac25d2d1f55835ded318c26004db85b70a3575b1)
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.