The command DGzero creates a zero biform, form, vector or tensor.  The degree of the form or the index type of the tensor is specified as a second argument.  This command is useful when, for example, a tensor T is to be computed recursively as sum of  tensors.

The command DGvolume creates the standard top-dimensional form on a manifold M or a biform of top degree horizontal on the jet space J^k(E),  where E is a fiber bundle over a base manifold M. It is convenient to use this command whenever a Lagrangian for some variational problem is to be defined as a biform of  top horizontal degree on the jet space.  The command DGvolume accepts as a second optional argument a Maple expression k, where DGvolume(keyword2,  k) = k  &mult DGvolume(keyword2).

The command DGscalar constructs a degree 0 form, a rank 0 tensor, or a biform of degree [0, 0].  

The command DGscalar accepts as a second optional argument a Maple expression k, where DGscalar(keyword3, k) = k &mult DGscalar(keyword3).

The command DGzero is part of the DifferentialGeometry:-Tools package, and so can be used in the form DGzero(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGzero.  The commands DGvolume and DGscalar work the same way.

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[p\,q\right]\,\left[u\right]\,E\,1\right)\:$

$\mathrm{DGinfo}\left(''CurrentFrame''\right)$

$E$

$\mathrm{DGzero}\left(''vector''\right)$

$0\mathrm{D\_p}$

$\mathrm{DGzero}\left(''vector''\,M\right)$

$0\mathrm{D\_x}$

$\mathrm{DGzero}\left(''form''\,2\right)$

$0\mathrm{dx}\bigwedge \mathrm{dy}$

$\mathrm{DGzero}\left(''tensor''\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right)$

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}$

$\mathrm{DGzero}\left(''tensor''\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\,E\right)$

$0\mathrm{D\_p}\mathrm{dp}\mathrm{dp}$

$\mathrm{DGzero}\left(''biform''\,\left[2\,2\right]\,E\right)$

$0\mathrm{Dp}\bigwedge \mathrm{Dq}\bigwedge \mathrm{Cu}\left[\right]\bigwedge {\mathrm{Cu}}_1$

$\mathrm{DGvolume}\left(''form''\,\exp \left(-x^2-y^2\right)\,M\right)$

${\ⅇ}^{-x^2-y^2}\mathrm{dx}\bigwedge \mathrm{dy}$

$f≔\mathrm{DGscalar}\left(''form''\,\ln \left(x\right)\right)$

$f≔\ln \left(x\right)$

$\mathrm{lprint}\left(f\right)$

_DG([["form", M, 0], [[[], ln(x)]]])