 Laplacian - Maple Help

Tensor[Laplacian] - find the Laplacian of a differential form with respect to a metric.

Calling Sequences

Laplacian(g, ${\mathbf{ω}}$)

Parameters

g      - a covariant metric tensor on an  n-dimensional manifold $M$

$\mathrm{ω}$      - a differential form on $M$ Description

 • 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

$\mathrm{δω}={\left(-1\right)}^{k}\ast d\ast \mathrm{ω}$ ,

where $\mathrm{ω}$ is a $p$-form, $k=\mathrm{np}+n+1$, and $n$ is the dimension of the underlying manifold $M$. The form $\mathrm{δω}$ has degree $p-1$. Since a metric tensor is needed to define the Hodge star operator * , a metric is also needed to define $\mathrm{δ}$.

 • The command Laplacian(g, ${\mathbf{ω}}$) computes the Laplacian of the differential form 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. Examples

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

Define a manifold $M$ with coordinates $\left(x,y,z\right)$ and a metric $g$.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 M > $g≔\mathrm{evalDG}\left(xz\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+{z}^{2}\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{≔}{x}{}{z}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{z}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.1)

Example 1.

Define a differential 1-form $\mathrm{α}$ and suppress the printing of the arguments of its coefficients with the PDEtools[declare] command.

 M > $\mathrm{\alpha }≔\mathrm{DGzip}\left(\left[a,b,c\right]\left(x,y,z\right),\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],"plus"\right)$
 ${\mathrm{\alpha }}{≔}{a}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{dx}}{+}{b}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{dy}}{+}{c}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{dz}}$ (2.2)
 M > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(\left[a,b,c\right]\left(x,y,z\right)\right)$
 ${a}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{a}$
 ${b}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{b}$
 ${c}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{c}$ (2.3)

Compute the Laplacian of $\mathrm{α}$.

 M > $\mathrm{map}\left(\mathrm{expand},\mathrm{Laplacian}\left(g,\mathrm{\alpha }\right)\right)$
 ${-}\left({-}\frac{{3}{}{{a}}_{{x}}}{{2}{}{z}{}{{x}}^{{2}}}{+}\frac{{{a}}_{{x}{,}{x}}}{{z}{}{x}}{+}\frac{{{c}}_{{x}}}{{z}}{+}\frac{{a}}{{z}{}{{x}}^{{3}}}{+}\frac{{{a}}_{{y}{,}{y}}}{{{z}}^{{2}}}{+}\frac{{{a}}_{{z}}}{{2}{}{z}}{+}{{a}}_{{z}{,}{z}}\right){}{\mathrm{dx}}{-}\left(\frac{{{b}}_{{y}{,}{y}}}{{{z}}^{{2}}}{+}\frac{{2}{}{{c}}_{{y}}}{{z}}{-}\frac{{{b}}_{{x}}}{{2}{}{z}{}{{x}}^{{2}}}{+}\frac{{{b}}_{{x}{,}{x}}}{{z}{}{x}}{-}\frac{{{b}}_{{z}}}{{2}{}{z}}{+}{{b}}_{{z}{,}{z}}\right){}{\mathrm{dy}}{-}\left(\frac{{a}}{{2}{}{{x}}^{{2}}{}{{z}}^{{2}}}{-}\frac{{{a}}_{{x}}}{{x}{}{{z}}^{{2}}}{+}\frac{{3}{}{{c}}_{{z}}}{{2}{}{z}}{-}\frac{{3}{}{c}}{{2}{}{{z}}^{{2}}}{+}{{c}}_{{z}{,}{z}}{-}\frac{{2}{}{{b}}_{{y}}}{{{z}}^{{3}}}{-}\frac{{{c}}_{{x}}}{{2}{}{{x}}^{{2}}{}{z}}{+}\frac{{{c}}_{{x}{,}{x}}}{{x}{}{z}}{+}\frac{{{c}}_{{y}{,}{y}}}{{{z}}^{{2}}}\right){}{\mathrm{dz}}$ (2.4)

Example 2.

Define a 2 form $\mathrm{β}$ and compute its Laplacian.

 M > $\mathrm{\beta }≔\mathrm{DGzip}\left(\left[a,b,c\right]\left(x,y,z\right),\left[\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy},\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz},\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right],"plus"\right)$
 ${\mathrm{\beta }}{≔}{a}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{b}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{+}{c}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.5)
 M > $\mathrm{map}\left(\mathrm{expand},\mathrm{Laplacian}\left(g,\mathrm{\beta }\right)\right)$
 ${-}\left({-}\frac{{3}{}{{a}}_{{x}}}{{2}{}{z}{}{{x}}^{{2}}}{+}\frac{{{a}}_{{x}{,}{x}}}{{z}{}{x}}{-}\frac{{{c}}_{{x}}}{{z}}{+}\frac{{a}}{{z}{}{{x}}^{{3}}}{+}\frac{{{a}}_{{y}{,}{y}}}{{{z}}^{{2}}}{+}\frac{{2}{}{{b}}_{{y}}}{{z}}{-}\frac{{3}{}{{a}}_{{z}}}{{2}{}{z}}{+}{{a}}_{{z}{,}{z}}\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}\left(\frac{{3}{}{{b}}_{{x}}}{{2}{}{z}{}{{x}}^{{2}}}{-}\frac{{{b}}_{{x}{,}{x}}}{{z}{}{x}}{-}\frac{{b}}{{z}{}{{x}}^{{3}}}{+}\frac{{2}{}{{a}}_{{y}}}{{{z}}^{{3}}}{+}\frac{{b}}{{2}{}{{z}}^{{2}}}{-}\frac{{{b}}_{{z}}}{{2}{}{z}}{-}{{b}}_{{z}{,}{z}}{-}\frac{{{b}}_{{y}{,}{y}}}{{{z}}^{{2}}}\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{+}\left({-}\frac{{{c}}_{{y}{,}{y}}}{{{z}}^{{2}}}{+}\frac{{a}}{{2}{}{{x}}^{{2}}{}{{z}}^{{2}}}{-}\frac{{{a}}_{{x}}}{{x}{}{{z}}^{{2}}}{-}\frac{{c}}{{2}{}{{z}}^{{2}}}{+}\frac{{{c}}_{{z}}}{{2}{}{z}}{-}{{c}}_{{z}{,}{z}}{+}\frac{{{c}}_{{x}}}{{2}{}{{x}}^{{2}}{}{z}}{-}\frac{{{c}}_{{x}{,}{x}}}{{x}{}{z}}\right){}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.6)

Example 3.

Compute the delta derivative of the 2 form $\mathrm{β}$.

 M > $\mathrm{Laplacian}:-\mathrm{ExteriorDerivativeStar}\left(g,\mathrm{\beta }\right)$
 $\frac{{2}{}{{a}}_{{y}}{+}{b}{}{z}{+}{2}{}{{z}}^{{2}}{}{{b}}_{{z}}}{{2}{}{{z}}^{{2}}}{}{\mathrm{dx}}{-}\frac{{-}{a}{+}{2}{}{x}{}{{a}}_{{x}}{+}{{x}}^{{2}}{}{c}{-}{2}{}{{x}}^{{2}}{}{{c}}_{{z}}{}{z}}{{2}{}{z}{}{{x}}^{{2}}}{}{\mathrm{dy}}{-}\frac{{-}{b}{}{z}{+}{2}{}{x}{}{{b}}_{{x}}{}{z}{+}{2}{}{{x}}^{{2}}{}{{c}}_{{y}}}{{2}{}{{z}}^{{2}}{}{{x}}^{{2}}}{}{\mathrm{dz}}$ (2.7)