If $g$ is a metric with components $g_{\mathrm{ij}}$, then $\mathrm{\ρ}\=\det {\left(g_{\mathrm{ij}}\right)}^{\frac{r}{2}}$ defines a scalar density of weight $r\.$

The program MetricDensity(g, r) returns the scalar density $\mathrm{\ρ}$.

By default, it is assumed that the metric $g$ has positive determinant. To calculate the proper metric density 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 MetricDensity(...) only after executing the commands with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-MetricDensity.

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

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

$\mathrm{g1}≔\mathrm{evalDG}\left(x\mathrm{dx}\&t\mathrm{dx}+y\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g1}:=x\mathrm{dx}\mathrm{dx}\+y\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$\mathrm{\ρ1}≔\mathrm{MetricDensity}\left(\mathrm{g1}\,1\right)$

$\mathrm{\ρ1}:=\sqrt{xy}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{\ρ1}\,''TensorDensityType''\right)$

$\left[\left[''bas''\,1\right]\right]$

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g2}:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{rho2a}≔\mathrm{MetricDensity}\left(\mathrm{g2}\,1\right)$

$\mathrm{rho2a}:=\mathrm{I}$

$\mathrm{rho2b}≔\mathrm{MetricDensity}\left(\mathrm{g2}\,1\,\mathrm{detmetric}=-1\right)$

$\mathrm{rho2b}:=1$

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

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

$\mathrm{g3}≔\mathrm{evalDG}\left(x\mathrm{du}\&t\mathrm{du}+y\mathrm{dv}\&t\mathrm{dv}+xy\mathrm{dw}\&t\mathrm{dw}\right)$

$\mathrm{g3}:=x\mathrm{du}\mathrm{du}\+y\mathrm{dv}\mathrm{dv}\+xy\mathrm{dw}\mathrm{dw}$

$\mathrm{\ρ3}≔\mathrm{MetricDensity}\left(\mathrm{g3}\,-1\right)$

$\mathrm{\ρ3}:=\frac{1}{\sqrt{x^2{y}^2}}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{\ρ3}\,''TensorDensityType''\right)$

$\left[\left[''vrt''\,-1\right]\right]$