Compte tenu de l'expression du tenseur métrique en coordonnées sphériques,
∇2f=1detg∂i(detggij∂jf){\displaystyle \nabla ^{2}f={\frac {1}{\sqrt {\det g}}}\partial _{i}\left({\sqrt {\det g}}\;g^{ij}\partial _{j}f\right)}
s'écrit
∇2f=(∂2∂r2+2r∂∂r+1r2∂2∂θ2+1r2tanθ∂∂θ+1r2sin2θ∂2∂ϕ2)f{\displaystyle \nabla ^{2}f=\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)f}
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