linguistic:translation
Three dimensions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In Cartesian coordinates, Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2. {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}

In cylindrical coordinates, Δ f = 1 ρ ∂ ∂ ρ (ρ ∂ f ∂ ρ) + 1 ρ 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2, {\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}},} where ρ {\displaystyle \rho } represents the radial distance, φ the azimuth angle and z the height.

In spherical coordinates: Δ f = 1 r 2 ∂ ∂ r (r 2 ∂ f ∂ r) + 1 r 2 sin θ ∂ ∂ θ (sin θ ∂ f ∂ θ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2, {\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} or Δ f = 1 r ∂ 2 ∂ r 2 (r f) + 1 r 2 sin θ ∂ ∂ θ (sin θ ∂ f ∂ θ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2, {\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(rf)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} by expanding the first and second term, these expressions read Δ f = ∂ 2 f ∂ r 2 + 2 r ∂ f ∂ r + 1 r 2 sin θ (cos θ ∂ f ∂ θ + sin θ ∂ 2 f ∂ θ 2) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2, {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}\sin \theta }}\left(\cos \theta {\frac {\partial f}{\partial \theta }}+\sin \theta {\frac {\partial ^{2}f}{\partial \theta ^{2}}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} where φ represents the azimuthal angle and θ the zenith angle or co-latitude.

In general curvilinear coordinates (ξ, ξ, ξ): Δ = ∇ ξ m ⋅ ∇ ξ n ∂ 2 ∂ ξ m ∂ ξ n + ∇ 2 ξ m ∂ ∂ ξ m = g m n (∂ 2 ∂ ξ m ∂ ξ n − Γ m n l ∂ ∂ ξ l), {\displaystyle \Delta =\nabla \xi ^{m}\cdot \nabla \xi ^{n}{\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}+\nabla ^{2}\xi ^{m}{\frac {\partial }{\partial \xi ^{m}}}=g^{mn}\left({\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}-\Gamma _{mn}^{l}{\frac {\partial }{\partial \xi ^{l}}}\right),}

where summation over the repeated indices is implied, g is the inverse metric tensor and Γ are the Christoffel symbols for the selected coordinates.