Let $T$ be a vector, a differential form, or a tensor which depends upon a number of parameters $\left\{f_1\, f_2 ..\.\, f_n\right\}$. These parameters may be constants or functions. Now let $\mathrm{\ℰ}$ be a differential-geometric construction depending upon $T$which can be implemented in Maple by a sequence of commands in the DifferentialGeometry package. For example, $T$ could be a metric tensor and $\mathrm{\ℰ}\mathit{ }$the Einstein tensor constructed from $g$. The command DGsolve will solve the equations obtained by setting to zero all the components of $\mathrm{\ℰ}\mathit{ }$for the unknowns $\left\{f_1\, f_2 ..\.\, f_n\right\}$. The output is a set containing those $T$ solving $ℰ$=0 (obtainable by Maple).

 Additional constraints (for example, initial conditions or inequalities) can be imposed upon the unknowns $\left\{f_1\, f_2 ..\.\, f_n\right\}$with the keyword argument auxiliaryequations.

The command DGsolve uses the general purpose solver PDEtools:-Solve to solve the system $\mathrm{\ℰ}\mathit{ }$=0 for the unknowns $\left\{f_1\, f_2 ..\.\, f_n\right\}$. The keyword argument method can be used to specify a particular Maple solver (for example, solve, pdsolve, dsolve) or a customized solver created by the user.

If the equations defined by $\mathrm{\ℰ}\mathit{ }$=0 are homogenous linear algebraic equations, then the command DGNullSpace can also be used.

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{du}\&s\mathrm{dv}+f\left(x\,u\right)\mathrm{du}\&t\mathrm{du}\right)$

$g:=\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+f\left(x\,u\right)\mathrm{du}\mathrm{du}\+\frac{1}{2}\mathrm{du}\mathrm{dv}\+\frac{1}{2}\mathrm{dv}\mathrm{du}$

$\mathrm{DGsolve}\left(\mathrm{EinsteinTensor}\left(g\right)\,g\right)$

$\left\{\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\left(\mathrm{\_F1}\left(u\right)x\+\mathrm{\_F2}\left(u\right)\right)\mathrm{du}\mathrm{du}\+\frac{1}{2}\mathrm{du}\mathrm{dv}\+\frac{1}{2}\mathrm{dv}\mathrm{du}\right\}$

$\mathrm{DGsolve}\left(\mathrm{BachTensor}\left(g\right)\,g\right)$

$\left\{\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\left(\frac{1}{6}\mathrm{\_F1}\left(u\right)x^3\+\frac{1}{2}\mathrm{\_F2}\left(u\right)x^2\+\mathrm{\_F3}\left(u\right)x\+\mathrm{\_F4}\left(u\right)\right)\mathrm{du}\mathrm{du}\+\frac{1}{2}\mathrm{du}\mathrm{dv}\+\frac{1}{2}\mathrm{dv}\mathrm{du}\right\}$

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

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

$\mathrm{\alpha }≔\mathrm{evalDG}\left(\mathrm{dx}\&w\mathrm{dy}+r\mathrm{dx}\&w\mathrm{du}+s\mathrm{dy}\&w\mathrm{dv}\right)\:$

$\mathrm{DGsolve}\left(\mathrm{\alpha }\&wedge\mathrm{\alpha }\,\mathrm{\alpha }\,\left\{r\,s\right\}\right)$

$\left\{\mathrm{dx}\bigwedge \mathrm{dy}\+r\mathrm{dx}\bigwedge \mathrm{du}\,\mathrm{dx}\bigwedge \mathrm{dy}\+s\mathrm{dy}\bigwedge \mathrm{dv}\right\}$

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

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

$\mathrm{Gamma}≔\mathrm{Connection}\left(-\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}+\left(\mathrm{D\_y}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)$

$\mathrm{\Γ}:=-\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\+\mathrm{D\_y}\mathrm{dy}\mathrm{dx}$

$C≔\left[\cos \left(t\right)\,\sin \left(t\right)\right]$

$C:=\left[\cos \left(t\right)\,\sin \left(t\right)\right]$

$X≔\mathrm{evalDG}\left(A\left(t\right)\mathrm{D\_x}+B\left(t\right)\mathrm{D\_y}\right)$

$X:=A\left(t\right)\mathrm{D\_x}\+B\left(t\right)\mathrm{D\_y}$

$\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C\,X\,\mathrm{Gamma}\,t\right)\,X\right)$

$\left\{\mathrm{\_C2}{\ⅇ}^{\sin \left(t\right)}\mathrm{D\_x}\+\mathrm{\_C1}{\ⅇ}^{-\cos \left(t\right)}\mathrm{D\_y}\right\}$

$\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C\,X\,\mathrm{Gamma}\,t\right)\,X\,\mathrm{auxiliaryequations}=\left\{A\left(0\right)=1\,B\left(0\right)=0\right\}\right)$

$\left\{{\ⅇ}^{\sin \left(t\right)}\mathrm{D\_x}\right\}$

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

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

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

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

$F≔\mathrm{evalDG}\left(A\left(x\,y\,z\,t\right)\mathrm{dx}\&w\mathrm{dy}+B\left(x\,y\,z\,t\right)\mathrm{dx}\&w\mathrm{dt}\right)$

$F:=A\left(x\,y\,z\,t\right)\mathrm{dx}\bigwedge \mathrm{dy}\+B\left(x\,y\,z\,t\right)\mathrm{dx}\bigwedge \mathrm{dt}$

$\mathrm{DGsolve}\left(\left[\mathrm{ExteriorDerivative}\left(F\right)\,\mathrm{ExteriorDerivative}\left(\mathrm{HodgeStar}\left(g\,F\,\mathrm{detmetric}=-1\right)\right)\right]\,F\right)$

$\left\{\left(\mathrm{\_F1}\left(t\+y\right)\+\mathrm{\_F2}\left(t-y\right)\right)\mathrm{dx}\bigwedge \mathrm{dy}\+\left(\mathrm{\_F1}\left(t\+y\right)-\mathrm{\_F2}\left(t-y\right)\+\mathrm{\_C1}\right)\mathrm{dx}\bigwedge \mathrm{dt}\right\}$