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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 :
.
Operator sačinjava Laplasov operator (
) i dvostruki izvod po vremenu :
![{\displaystyle \Box =\Delta -{\frac {\partial ^{2}}{c^{2}\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/853b0c13a6686c6213b8156b40c998bc0b2d8468)
U teoriji relativnosti, koristi se zapis sa Ajnštajnovim indeksima.
.
gde je kovarijantni zapis,
![{\displaystyle \partial ^{\mu }=\eta ^{\mu \nu }\partial _{\nu }=\left(\partial _{ct},-\partial _{x},-\partial _{y},-\partial _{z}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3393eb716f500303970f8a375e017251c8f89b06)
Proizvod je definisan kao d'Alamberov operator.
![{\displaystyle \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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60dd4b7c5a1f6d60c52f02f489a0663ca28b6f54)
U različitim koordinatnim sistemima[uredi | uredi izvor]
D'Alamberov operator u sfernim koordinatama:
![{\displaystyle {\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}}};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c855dcb932a37e5add941f6093d171cd69eebaef)
D'Alamberov operator u cilindričnim koordinatama:
![{\displaystyle {\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}}};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cca016f461861f380a1bc05163f04cb1d42f6d9b)
U opštim krivolinijskim koordinatama:
![{\displaystyle \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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93b5335ec22f4fc0d5417ae225a14e91ed3d3107)
gde je
metrički tenzor, a g je determinanta toga tenzora.