The Laplace-Beltrami operator $\mathrm{\Δ}$ is the second order linear differential operator which acts on $p$-forms $\mathrm{\ω}$ by

The $\mathrm{\δ}$ differential operator is the first-order linear differential operator defined in terms of the exterior derivative operator $d$ and the Hodge star operator * by

The command Laplacian(g, $\mathbf{\ω}$) computes the Laplacian of the differential form $\mathrm{\ω}$with respect to the metric tensor $g$.

The command Laplacian:-ExteriorDerivativeStar(g, $\mathbf{\ω}$) computes $\mathrm{\δ}$ applied to $\mathrm{\ω}$.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form Laplacian(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-Laplacian.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$g≔\mathrm{evalDG}\left(xz\mathrm{dx}\&t\mathrm{dx}+z^2\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔xz\mathrm{dx}\mathrm{dx}+z^2\mathrm{dy}\mathrm{dy}+\mathrm{dz}\mathrm{dz}$

$\mathrm{\alpha }≔\mathrm{DGzip}\left(\left[a\,b\,c\right]\left(x\,y\,z\right)\,\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]\,''plus''\right)$

$\mathrm{\alpha }≔a\left(x\,y\,z\right)\mathrm{dx}+b\left(x\,y\,z\right)\mathrm{dy}+c\left(x\,y\,z\right)\mathrm{dz}$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(\left[a\,b\,c\right]\left(x\,y\,z\right)\right)$

$a\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}a$

$b\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}b$

$c\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}c$

$\mathrm{map}\left(\mathrm{expand}\,\mathrm{Laplacian}\left(g\,\mathrm{\alpha }\right)\right)$

$-\left(-\frac{3a_x}{2z{x}^2}+\frac{a_{x,x}}{zx}+\frac{c_x}{z}+\frac{a}{z{x}^3}+\frac{a_{y,y}}{z^2}+\frac{a_z}{2z}+a_{z,z}\right)\mathrm{dx}-\left(\frac{b_{y,y}}{z^2}+\frac{2c_y}{z}-\frac{b_x}{2z{x}^2}+\frac{b_{x,x}}{zx}-\frac{b_z}{2z}+b_{z,z}\right)\mathrm{dy}-\left(\frac{a}{2x^2{z}^2}-\frac{a_x}{x{z}^2}+\frac{3c_z}{2z}-\frac{3c}{2z^2}+c_{z,z}-\frac{2b_y}{z^3}-\frac{c_x}{2x^{2}z}+\frac{c_{x,x}}{xz}+\frac{c_{y,y}}{z^2}\right)\mathrm{dz}$

$\mathrm{\beta }≔\mathrm{DGzip}\left(\left[a\,b\,c\right]\left(x\,y\,z\right)\,\left[\mathrm{dx}\&wedge\mathrm{dy}\,\mathrm{dx}\&wedge\mathrm{dz}\,\mathrm{dy}\&wedge\mathrm{dz}\right]\,''plus''\right)$

$\mathrm{\beta }≔a\mathrm{dx}\bigwedge \mathrm{dy}+b\mathrm{dx}\bigwedge \mathrm{dz}+c\mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{map}\left(\mathrm{expand}\,\mathrm{Laplacian}\left(g\,\mathrm{\beta }\right)\right)$

$-\left(-\frac{3a_x}{2z{x}^2}+\frac{a_{x,x}}{zx}-\frac{c_x}{z}+\frac{a}{z{x}^3}+\frac{a_{y,y}}{z^2}+\frac{2b_y}{z}-\frac{3a_z}{2z}+a_{z,z}\right)\mathrm{dx}\bigwedge \mathrm{dy}+\left(\frac{3b_x}{2z{x}^2}-\frac{b_{x,x}}{zx}-\frac{b}{z{x}^3}+\frac{2a_y}{z^3}+\frac{b}{2z^2}-\frac{b_z}{2z}-b_{z,z}-\frac{b_{y,y}}{z^2}\right)\mathrm{dx}\bigwedge \mathrm{dz}+\left(-\frac{c_{y,y}}{z^2}+\frac{a}{2x^2{z}^2}-\frac{a_x}{x{z}^2}-\frac{c}{2z^2}+\frac{c_z}{2z}-c_{z,z}+\frac{c_x}{2x^{2}z}-\frac{c_{x,x}}{xz}\right)\mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{Laplacian}:-\mathrm{ExteriorDerivativeStar}\left(g\,\mathrm{\beta }\right)$

$\frac{2a_y+bz+2z^2{b}_z}{2z^2}\mathrm{dx}-\frac{-a+2x{a}_x+x^{2}c-2x^2{c}_{z}z}{2z{x}^2}\mathrm{dy}-\frac{-bz+2x{b}_{x}z+2x^2{c}_y}{2z^2{x}^2}\mathrm{dz}$