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