Let $g$ be a metric with signature $\left(1\,-1\,-1\,-1\right)$ and $\left(L\,N\,M\,\bar{M}\right)$be a null tetrad for $g$. The Newman-Penrose directional derivative operators take derivatives of functions in the directions of the 4 vectors of the tetrad. See Newman and Penrose, Stewart.

The NPDirectionalDerivatives command returns a table with 4 entries "D", "Delta", "delta", "barDelta" corresponding to the customary labels assigned to the NP directional derivative operators $\(D\,\mathrm{\Δ}\,$δ, $\overline{\mathrm{\δ}}$). With optional keyword argument output = "sequence", the directional derivative operators are returned as a sequence of 4 Maple procedures.

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

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

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

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

$g≔\mathrm{evalDG}\left(x^2\mathrm{dt}\&t\mathrm{dt}-y^2\mathrm{dx}\&t\mathrm{dx}-z^2\mathrm{dy}\&t\mathrm{dy}-t^2\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=x^2\mathrm{dt}\mathrm{dt}-y^2\mathrm{dx}\mathrm{dx}-z^2\mathrm{dy}\mathrm{dy}-t^2\mathrm{dz}\mathrm{dz}$

$\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D\_t}+\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D\_z}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D\_t}-\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D\_z}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D\_x}+\frac{\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}}{z}\mathrm{D\_y}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D\_x}-\frac{\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}}{z}\mathrm{D\_y}\right]\right)$

$\mathrm{NTetrad}:=\left[\frac{\sqrt{2}}{2x}\mathrm{D\_t}+\frac{\sqrt{2}}{2t}\mathrm{D\_z}\,\frac{\sqrt{2}}{2x}\mathrm{D\_t}-\frac{\sqrt{2}}{2t}\mathrm{D\_z}\,\frac{\sqrt{2}}{2y}\mathrm{D\_x}+\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{z}\mathrm{D\_y}\,\frac{\sqrt{2}}{2y}\mathrm{D\_x}-\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{z}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{NTetrad}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NPDirDer}≔\mathrm{NPDirectionalDerivatives}\left(\mathrm{NTetrad}\right)$

$\mathrm{NPDirDer}:=\mathrm{table}\left(\left[''Delta''\=\left(f\→\mathrm{LieDerivative}\left(E_2\,f\right)\right)\,''delta''\=\left(f\→\mathrm{LieDerivative}\left(E_3\,f\right)\right)\,''D''\=\left(f\→\mathrm{LieDerivative}\left(E_1\,f\right)\right)\,''bardelta''\=\left(f\→\mathrm{LieDerivative}\left(E_4\,f\right)\right)\right]\right)$

$\mathrm{indices}\left(\mathrm{NPDirDer}\right)$

$\left[''Delta''\right]\,\left[''delta''\right]\,\left[''D''\right]\,\left[''bardelta''\right]$

$X≔\mathrm{NPDirDer}\left[''D''\right]$

$X:={\mathrm{NPDirDer}}_{''D''}$

$X\left(f\left(t\,x\,y\,z\right)\right)$

$\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial t}f\left(t\,x\,y\,z\right)\right)}{x}\+\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial z}f\left(t\,x\,y\,z\right)\right)}{t}$

$\mathrm{LieDerivative}\left(\mathrm{NTetrad}\left[1\right]\,f\left(t\,x\,y\,z\right)\right)$

$\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial t}f\left(t\,x\,y\,z\right)\right)}{x}\+\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial z}f\left(t\,x\,y\,z\right)\right)}{t}$

$d,\mathrm{\Delta },\mathrm{\delta },\mathrm{bardela}≔\mathrm{NPDirectionalDerivatives}\left(\mathrm{NTetrad}\,\mathrm{output}=''Sequence''\right)$

$d\,\mathrm{\Δ}\,\mathrm{\δ}\,\mathrm{bardela}:=f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_1\,f\right)\,f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_2\,f\right)\,f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_3\,f\right)\,f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_4\,f\right)$

$\mathrm{\Delta }\left(f\left(t\,x\,y\,z\right)\right)$

$\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial t}f\left(t\,x\,y\,z\right)\right)}{x}-\frac{1}{2}\frac{\sqrt{2}\left(\frac{\partial }{\partial z}f\left(t\,x\,y\,z\right)\right)}{t}$

$\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{NTetrad}\,\mathrm{NP}\right)$

$\mathrm{FD}:=\left[\left[\mathrm{E1}\,\mathrm{E2}\right]\=\frac{1}{2}\frac{\sqrt{2}\mathrm{E1}}{tx}-\frac{1}{2}\frac{\sqrt{2}\mathrm{E2}}{tx}\,\left[\mathrm{E1}\,\mathrm{E3}\right]\=\frac{1}{4}\frac{\sqrt{2}\mathrm{E1}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E2}}{xy}-\frac{1}{4}\frac{\sqrt{2}\mathrm{E3}}{tz}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E4}}{tz}\,\left[\mathrm{E1}\,\mathrm{E4}\right]\=\frac{1}{4}\frac{\sqrt{2}\mathrm{E1}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E2}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E3}}{tz}-\frac{1}{4}\frac{\sqrt{2}\mathrm{E4}}{tz}\,\left[\mathrm{E2}\,\mathrm{E3}\right]\=\frac{1}{4}\frac{\sqrt{2}\mathrm{E1}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E2}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E3}}{tz}-\frac{1}{4}\frac{\sqrt{2}\mathrm{E4}}{tz}\,\left[\mathrm{E2}\,\mathrm{E4}\right]\=\frac{1}{4}\frac{\sqrt{2}\mathrm{E1}}{xy}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E2}}{xy}-\frac{1}{4}\frac{\sqrt{2}\mathrm{E3}}{tz}\+\frac{1}{4}\frac{\sqrt{2}\mathrm{E4}}{tz}\,\left[\mathrm{E3}\,\mathrm{E4}\right]\=-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{E3}}{zy}-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{E4}}{zy}\right]$

$\mathrm{DGsetup}\left(\mathrm{FD}\right)$

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

$\mathrm{NPDirectionalDerivatives}\left(\mathrm{NP}\right)$

$\mathrm{table}\left(\left[''Delta''\=\left(f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_2\,f\right)\right)\,''delta''\=\left(f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_3\,f\right)\right)\,''D''\=\left(f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_1\,f\right)\right)\,''bardelta''\=\left(f\→\mathrm{DifferentialGeometry:-LieDerivative}\left(E_4\,f\right)\right)\right]\right)$