D'Alamberov operator

D'Alamberov operator je diferencijalni operator drugog reda. D'Alamberov operator je u stvari Laplasov operator u prostoru Minkovskog - (t, x, y, z). Nazvan je po francuskom matematičaru Žan le Ron d'Alamberu.

Operator se često koristi u fizici elektromagnetskog polja - talasna jednačina svetla. Oznaka za d'Alamberov operator je kvadrat : $\Box$.

Operator sačinjava Laplasov operator ($\Delta$) i dvostruki izvod po vremenu :

$\Box =\Delta -{\frac {\partial ^{2}}{c^{2}\partial t^{2}}}$

Ajnštajnov zapis

U teoriji relativnosti, koristi se zapis sa Ajnštajnovim indeksima.

$\partial _{\mu }=\left(\partial _{ct},\nabla \right)=\left(\partial _{ct},\partial _{x},\partial _{y},\partial _{z}\right)$.

gde je kovarijantni zapis,

$\partial ^{\mu }=\eta ^{\mu \nu }\partial _{\nu }=\left(\partial _{ct},-\partial _{x},-\partial _{y},-\partial _{z}\right)$

Proizvod je definisan kao d'Alamberov operator.

$\Box :=\partial ^{\mu }\partial _{\mu }={\frac {\partial ^{2}}{c^{2}\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}={\frac {\partial ^{2}}{c^{2}\partial t^{2}}}-\Delta$

U različitim koordinatnim sistemima

D'Alamberov operator u sfernim koordinatama:

${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}\sin ^{2}\Theta }}{\frac {\partial }{\partial \Theta }}\left(\sin \Theta {\frac {\partial u}{\partial \Theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\Theta }}{\frac {\partial ^{2}u}{\partial \varphi ^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}};$

D'Alamberov operator u cilindričnim koordinatama:

${\frac {1}{\rho ^{2}}}{\frac {\partial }{\partial \rho }}\left(\rho ^{2}{\frac {\partial u}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}u}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}};$

U opštim krivolinijskim koordinatama:

$\square u\equiv {\frac {1}{\sqrt {-g}}}{\frac {\partial }{\partial x^{\nu }}}\left({\sqrt {-g}}\,g^{\mu \nu }{\frac {\partial u}{\partial x^{\mu }}}\right),$

gde je $g_{\mu \nu }$ metrički tenzor, a g je determinanta toga tenzora.