apply the Hodge star operator to a differential form - Maple Programming Help

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Tensor[HodgeStar] - apply the Hodge star operator to a differential form

Calling Sequences

HodgeStar(g, omega)

Parameters

g      - a metric tensor

omega  - a differential form

option - (optional) the keyword argument detmetric

Examples

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

Example 1.

First create a 5-dimensional manifold $M$ and define a metric tensor $g$ on the tangent space of $M$.

 E > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}\right],\mathrm{M1}\right):$
 M1 > $g≔\mathrm{evalDG}\left(\mathrm{dx1}&t\mathrm{dx1}+\mathrm{dx2}&t\mathrm{dx2}+\mathrm{dx3}&t\mathrm{dx3}+\mathrm{dx4}&t\mathrm{dx4}+\mathrm{dx5}&t\mathrm{dx5}\right)$
 ${g}{:=}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}{+}{\mathrm{dx5}}{}{\mathrm{dx5}}$ (1.1)

The standard basis  is an orthonormal basis for $g$ and therefore the Hodge star is easily computed.

 M1 > $\mathrm{HodgeStar}\left(g,\mathrm{dx1}\right)$
 ${\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}$ (1.2)
 M1 > $\mathrm{HodgeStar}\left(g,\mathrm{dx2}\right)$
 ${-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}$ (1.3)
 M1 > $\mathrm{HodgeStar}\left(g,\mathrm{dx2}&w\mathrm{dx3}\right)$
 ${\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}$ (1.4)
 M1 > $\mathrm{HodgeStar}\left(g,\mathrm{dx2}&w\mathrm{dx4}\right)$
 ${-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx5}}$ (1.5)
 M1 > $\mathrm{HodgeStar}\left(g,\left(\mathrm{dx2}&w\mathrm{dx3}\right)&w\mathrm{dx4}\right)$
 ${-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx5}}$ (1.6)

Example 2.

To show the dependence of the Hodge star upon the metric, we consider a general metric $g$ on a 2-dimensional manifold.

 M1 > $\mathrm{DGsetup}\left(\left[x,y\right],\mathrm{M2}\right):$
 M2 > $g≔\mathrm{evalDG}\left(a\mathrm{dx}&t\mathrm{dx}+b\left(\mathrm{dx}&t\mathrm{dy}+\mathrm{dy}&t\mathrm{dx}\right)+c\mathrm{dy}&t\mathrm{dy}\right)$
 ${g}{:=}{a}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{b}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{b}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{c}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (1.7)
 M2 > $\mathrm{HodgeStar}\left(g,\mathrm{dx}\right)$
 $\sqrt{\frac{{1}}{{a}{}{c}{-}{{b}}^{{2}}}}{}{b}{}{\mathrm{dx}}{+}\sqrt{\frac{{1}}{{a}{}{c}{-}{{b}}^{{2}}}}{}{c}{}{\mathrm{dy}}$ (1.8)
 M2 > $\mathrm{HodgeStar}\left(g,\mathrm{dy}\right)$
 ${-}\sqrt{\frac{{1}}{{a}{}{c}{-}{{b}}^{{2}}}}{}{a}{}{\mathrm{dx}}{-}\sqrt{\frac{{1}}{{a}{}{c}{-}{{b}}^{{2}}}}{}{b}{}{\mathrm{dy}}$ (1.9)
 M2 > $f≔\mathrm{HodgeStar}\left(g,\mathrm{dx}&w\mathrm{dy}\right)$
 ${f}{:=}\sqrt{\frac{{1}}{{a}{}{c}{-}{{b}}^{{2}}}}$ (1.10)
 M2 > $\mathrm{HodgeStar}\left(g,f\right)$
 ${\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}$ (1.11)

Example 3.

The Laplacian of a function with respect to a metric $g$ can be calculated using the exterior derivative operation and the Hodge star operator.

To illustrate this result, we use the Euclidean metric in polar coordinates $\left(r,\mathrm{ϑ}\right)$.

 M2 > $\mathrm{DGsetup}\left(\left[r,\mathrm{θ}\right],\mathrm{M3}\right):$
 M3 > $g≔\mathrm{evalDG}\left(\mathrm{dr}&t\mathrm{dr}+{r}^{2}\mathrm{dtheta}&t\mathrm{dtheta}\right)$
 ${g}{:=}{\mathrm{dr}}{}{\mathrm{dr}}{+}{{r}}^{{2}}{}{\mathrm{dtheta}}{}{\mathrm{dtheta}}$ (1.12)

To simplify the definition of the Laplacian, we define the Hodge operator with $g$ fixed.

 M3 > $\mathrm{Hodge}≔f→\mathrm{HodgeStar}\left(g,f\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0
 ${\mathrm{Hodge}}{:=}{f}{→}{\mathrm{DifferentialGeometry:-Tensor}}{:-}{\mathrm{HodgeStar}}{}\left({g}{,}{f}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{assuming}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{r}$ (1.13)

To display the Laplacian of $\mathrm{φ}$ in compact form we invoke the PDEtools[declare] command.

 M3 > $\mathrm{PDEtools}[\mathrm{declare}]\left(\mathrm{φ}\left(r,\mathrm{θ}\right)\right)$
 ${\mathrm{φ}}{}\left({r}{,}{\mathrm{θ}}\right){}{\mathrm{will now be displayed as}}{}{\mathrm{φ}}$ (1.14)

Here is the formula for the Laplacian in terms of HodgeStar and ExteriorDerivative.  Recall that @ is the composition of functions.

 M3 > $\mathrm{Δ}≔\left(\left(\mathrm{Hodge}@\mathrm{ExteriorDerivative}\right)@\mathrm{Hodge}\right)@\mathrm{ExteriorDerivative}\left(\mathrm{φ}\left(r,\mathrm{θ}\right)\right)$
 ${\mathrm{Δ}}{:=}\frac{{r}{}{{\mathrm{φ}}}_{{r}}{+}{{r}}^{{2}}{}{{\mathrm{φ}}}_{{r}{,}{r}}{+}{{\mathrm{φ}}}_{{\mathrm{θ}}{,}{\mathrm{θ}}}}{{{r}}^{{2}}}$ (1.15)

Example 4.

The HodgeStar program also works in the more general context of a vector bundle $E\to M$.

 M3 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v,w\right],E\right)$
 ${\mathrm{frame name: E}}$ (1.16)
 E > $g≔\mathrm{evalDG}\left(\mathrm{du}&t\mathrm{du}+\mathrm{dv}&t\mathrm{dv}+\mathrm{dw}&t\mathrm{dw}\right)$
 ${g}{:=}{\mathrm{du}}{}{\mathrm{du}}{+}{\mathrm{dv}}{}{\mathrm{dv}}{+}{\mathrm{dw}}{}{\mathrm{dw}}$ (1.17)
 E > $\mathrm{HodgeStar}\left(g,\mathrm{du}&w\mathrm{dv}-3\mathrm{du}&w\mathrm{dw}+2\mathrm{dv}&w\mathrm{dw}\right)$
 ${2}{}{\mathrm{du}}{+}{3}{}{\mathrm{dv}}{+}{\mathrm{dw}}$ (1.18)

Example 5.

The HodgeStar operation can also be performed using an indefinite metric. The keyword argument detmetric = -1 must be used when the metric has negative determinant.

 E > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],\mathrm{M5}\right):$
 M1 > $g≔\mathrm{evalDG}\left(\mathrm{dx1}&t\mathrm{dx1}+\mathrm{dx2}&t\mathrm{dx2}+\mathrm{dx3}&t\mathrm{dx3}-\mathrm{dx4}&t\mathrm{dx4}\right)$
 ${g}{:=}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{-}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (1.19)

 M1 > $\mathrm{HodgeStar}\left(g,\mathrm{dx1},\mathrm{detmetric}=-1\right)$
 ${\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}$ (1.20)
 M5 > $\mathrm{HodgeStar}\left(g,\mathrm{dx3}&w\mathrm{dx4},\mathrm{detmetric}=-1\right)$
 ${-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}$ (1.21)

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