The Hodge star operator is customarily denoted by $\ast$ and maps $p$-forms on an $n$-dimensional manifold to $\left(n-p\right)$-forms. To define the Hodge star operator, let ${\mathrm{\&omega;}}_1\&comma;{\mathrm{\&omega;}}_2\&comma;..\&period;\&comma;{\mathrm{\&omega;}}_n$ be an orthonormal basis of 1-forms with respect to the metric $g$. Pick integers $i_1\&comma;i_2\&comma;..\&period;\&comma;i_r$ with $1\le i_1<i_2..\&period;<i_r\le n$ and let $1\le j_1<j_2<..\&period;<j_s\le n$ be the complementary set of integers $\left(r\&equals;n-s\right)$.  Let $\mathrm{\&omega;}\&equals;{\mathrm{\&omega;}}_{i_1}\wedge {\mathrm{\&omega;}}_{i_2}\wedge ..\&period; \wedge {\mathrm{\&omega;}}_{i_r}$. Then

The command HodgeStar(g, omega) returns the Hodge star of the form omega with respect to the metric g.

The command HodgeStar(g, omega) calculates the Hodge star of the form omega as follows. [i] Convert omega to skew-symmetric, covariant rank r tensor T. [ii] Fully contract T with the contravariant permutation symbol and multiply by the weight 1 scalar density defined by the metric g. The result is a skew-symmetric contravariant rank n-r tensor S. [iii] Lower all the indices of S with the metric g and convert the resulting skew-symmetric, contravariant rank n-r tensor to a differential form.

By default, it is assumed that the metric g has positive determinant. To calculate the Hodge star operation with respect to a metric with negative determinant, include the keyword argument detmetric = -1.

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

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\&colon;$$\mathrm{with}\left(\mathrm{Tensor}\right)\&colon;$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\&comma;\mathrm{x5}\right]\&comma;\mathrm{M1}\right)\&colon;$

$g≔\mathrm{evalDG}\left(\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}+\mathrm{dx5}\&t\mathrm{dx5}\right)$

$g:=\mathrm{dx1}\mathrm{dx1}\&plus;\mathrm{dx2}\mathrm{dx2}\&plus;\mathrm{dx3}\mathrm{dx3}\&plus;\mathrm{dx4}\mathrm{dx4}\&plus;\mathrm{dx5}\mathrm{dx5}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx1}\right)$

$\mathrm{dx2}\bigwedge \mathrm{dx3}\bigwedge \mathrm{dx4}\bigwedge \mathrm{dx5}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx2}\right)$

$-\mathrm{dx1}\bigwedge \mathrm{dx3}\bigwedge \mathrm{dx4}\bigwedge \mathrm{dx5}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx2}\&w\mathrm{dx3}\right)$

$\mathrm{dx1}\bigwedge \mathrm{dx4}\bigwedge \mathrm{dx5}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx2}\&w\mathrm{dx4}\right)$

$-\mathrm{dx1}\bigwedge \mathrm{dx3}\bigwedge \mathrm{dx5}$

$\mathrm{HodgeStar}\left(g\&comma;\left(\mathrm{dx2}\&w\mathrm{dx3}\right)\&w\mathrm{dx4}\right)$

$-\mathrm{dx1}\bigwedge \mathrm{dx5}$

$\mathrm{DGsetup}\left(\left[x\&comma;y\right]\&comma;\mathrm{M2}\right)\&colon;$

$g≔\mathrm{evalDG}\left(a\mathrm{dx}\&t\mathrm{dx}+b\left(\mathrm{dx}\&t\mathrm{dy}+\mathrm{dy}\&t\mathrm{dx}\right)+c\mathrm{dy}\&t\mathrm{dy}\right)$

$g:=a\mathrm{dx}\mathrm{dx}\&plus;b\mathrm{dx}\mathrm{dy}\&plus;b\mathrm{dy}\mathrm{dx}\&plus;c\mathrm{dy}\mathrm{dy}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx}\right)$

$\sqrt{\frac{1}{ac-b^2}}b\mathrm{dx}\&plus;\sqrt{\frac{1}{ac-b^2}}c\mathrm{dy}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dy}\right)$

$-\sqrt{\frac{1}{ac-b^2}}a\mathrm{dx}-\sqrt{\frac{1}{ac-b^2}}b\mathrm{dy}$

$f≔\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx}\&w\mathrm{dy}\right)$

$f:=\sqrt{\frac{1}{ac-b^2}}$

$\mathrm{HodgeStar}\left(g\&comma;f\right)$

$\mathrm{dx}\bigwedge \mathrm{dy}$

$\mathrm{DGsetup}\left(\left[r\&comma;\mathrm{\theta }\right]\&comma;\mathrm{M3}\right)\&colon;$

$g≔\mathrm{evalDG}\left(\mathrm{dr}\&t\mathrm{dr}+r^2\mathrm{dtheta}\&t\mathrm{dtheta}\right)$

$g:=\mathrm{dr}\mathrm{dr}\&plus;r^2\mathrm{dtheta}\mathrm{dtheta}$

$\mathrm{Hodge}≔f\mapsto \left(\mathrm{HodgeStar}\left(g\&comma;f\right)\mathbf{assuming}0<r\right)$

$\mathrm{Hodge}:=f\&rarr;\mathrm{DifferentialGeometry:-Tensor}:-\mathrm{HodgeStar}\left(g\&comma;f\right)assuming0<r$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(\mathrm{\phi }\left(r\&comma;\mathrm{\theta }\right)\right)$

$\mathrm{\&phi;}\left(r\&comma;\mathrm{\&theta;}\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\&phi;}$

$\mathrm{\Delta }≔\left(\mathrm{Hodge}@\mathrm{ExteriorDerivative}@\mathrm{Hodge}@\mathrm{ExteriorDerivative}\right)\left(\mathrm{\phi }\left(r\&comma;\mathrm{\theta }\right)\right)$

$\mathrm{\&Delta;}:=\frac{r{\mathrm{\&phi;}}_r\&plus;r^2{\mathrm{\&phi;}}_{r\&comma;r}\&plus;{\mathrm{\&phi;}}_{\mathrm{\&theta;}\&comma;\mathrm{\&theta;}}}{r^2}$

$\mathrm{DGsetup}\left(\left[x\&comma;y\right]\&comma;\left[u\&comma;v\&comma;w\right]\&comma;E\right)$

$\mathrm{frame\; name:\; E}$

$g≔\mathrm{evalDG}\left(\mathrm{du}\&t\mathrm{du}+\mathrm{dv}\&t\mathrm{dv}+\mathrm{dw}\&t\mathrm{dw}\right)$

$g:=\mathrm{du}\mathrm{du}\&plus;\mathrm{dv}\mathrm{dv}\&plus;\mathrm{dw}\mathrm{dw}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{du}\&w\mathrm{dv}-3\mathrm{du}\&w\mathrm{dw}+2\mathrm{dv}\&w\mathrm{dw}\right)$

$2\mathrm{du}\&plus;3\mathrm{dv}\&plus;\mathrm{dw}$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right]\&comma;\mathrm{M5}\right)\&colon;$

$g≔\mathrm{evalDG}\left(\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}-\mathrm{dx4}\&t\mathrm{dx4}\right)$

$g:=\mathrm{dx1}\mathrm{dx1}\&plus;\mathrm{dx2}\mathrm{dx2}\&plus;\mathrm{dx3}\mathrm{dx3}-\mathrm{dx4}\mathrm{dx4}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx1}\&comma;\mathrm{detmetric}=-1\right)$

$\mathrm{dx2}\bigwedge \mathrm{dx3}\bigwedge \mathrm{dx4}$

$\mathrm{HodgeStar}\left(g\&comma;\mathrm{dx3}\&w\mathrm{dx4}\&comma;\mathrm{detmetric}=-1\right)$

$-\mathrm{dx1}\bigwedge \mathrm{dx2}$