DGsolve - Maple Help

DifferentialGeometry[DGsolve] - solve a list of tensor equations for an unknown list of tensors

 Calling Sequence DGsolve(Eq, T, options)

Parameters

 Eq - a vector, differential form or tensor constructed from the objects in the 2nd argument; or list of such. The vanishing of these tensors defines the equations to be solved. T - a vector, differential form, or tensor, depending upon a number of arbitrary parameters or functions; or a list of such auxiliaryequations - (optional) a keyword argument to specify a set of auxiliary equations, to be solved in conjunction with the equations specified by the first argument unknowns - (optional) list of parameters and functions, explicitly specifying the unknowns to be solved for. method - (optional) a Maple procedure which will be used to solve the equations other - (optional) additional arguments to be passed to the procedure used the solve the equations

Description

 • Let  be a vector, a differential form, or a tensor which depends upon a number of parameters . These parameters may be constants or functions. Now let $\mathrm{ℰ}$ be a differential-geometric construction depending upon which can be implemented in Maple by a sequence of commands in the DifferentialGeometry package. For example, $T$ could be a metric tensor and the Einstein tensor constructed from $g$. The command DGsolve will solve the equations obtained by setting to zero all the components of for the unknowns . 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 with the keyword argument auxiliaryequations.
 • The command DGsolve uses the general purpose solver PDEtools:-Solve to solve the system =0 for the unknowns . 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 =0 are homogenous linear algebraic equations, then the command DGNullSpace can also be used.

Examples

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

Example 1.

Let  be a 4-dimensional space. We define a metric tensor depending upon an arbitrary function. We find the metrics which have vanishing Einstein tensor, and vanishing Bach tensor.

 > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.1)
 > $gâ‰”\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{du}&s\mathrm{dv}+f\left(x,u\right)\mathrm{du}&t\mathrm{du}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{f}{}\left({x}{,}{u}\right){}{\mathrm{du}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}$ (4.2)

Here are the metrics of the form (4.2) with vanishing Einstein tensor.

 M > $\mathrm{DGsolve}\left(\mathrm{EinsteinTensor}\left(g\right),g\right)$
 $\left\{{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}\left({\mathrm{_F1}}{}\left({u}\right){}{x}{+}{\mathrm{_F2}}{}\left({u}\right)\right){}{\mathrm{du}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}\right\}$ (4.3)

Here are the metrics of the form (4.2) with vanishing Bach tensor.

 M > $\mathrm{DGsolve}\left(\mathrm{BachTensor}\left(g\right),g\right)$
 $\left\{{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}\left(\frac{{1}}{{6}}{}{\mathrm{_F1}}{}\left({u}\right){}{{x}}^{{3}}{+}\frac{{1}}{{2}}{}{\mathrm{_F2}}{}\left({u}\right){}{{x}}^{{2}}{+}{\mathrm{_F3}}{}\left({u}\right){}{x}{+}{\mathrm{_F4}}{}\left({u}\right)\right){}{\mathrm{du}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{\mathrm{dv}}{}{\mathrm{du}}\right\}$ (4.4)

Example 2.

In this example we define a 2-form which depends upon parameters . We find those values of the parameters for which

 M > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.5)
 M > $\mathrm{α}â‰”\mathrm{evalDG}\left(\mathrm{dx}&w\mathrm{dy}+r\mathrm{dx}&w\mathrm{du}+s\mathrm{dy}&w\mathrm{dv}\right):$
 M > $\mathrm{DGsolve}\left(\mathrm{α}&wedge\mathrm{α},\mathrm{α},\left\{r,s\right\}\right)$
 $\left\{{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{r}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{du}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{s}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dv}}\right\}$ (4.6)

Example 3.

We define a connection $\mathrm{Γ}$ and calculate the parallel transport of a vector $X\left(t\right)$ along a curve $C\left(t\right)$.

 M > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.7)
 M > $\mathrm{Gamma}â‰”\mathrm{Connection}\left(-\left(\mathrm{D_x}&t\mathrm{dx}\right)&t\mathrm{dy}+\left(\mathrm{D_y}&t\mathrm{dy}\right)&t\mathrm{dx}\right)$
 ${\mathrm{Γ}}{:=}{-}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (4.8)
 M > $Câ‰”\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]$
 ${C}{:=}\left[{\mathrm{cos}}{}\left({t}\right){,}{\mathrm{sin}}{}\left({t}\right)\right]$ (4.9)
 M > $Xâ‰”\mathrm{evalDG}\left(A\left(t\right)\mathrm{D_x}+B\left(t\right)\mathrm{D_y}\right)$
 ${X}{:=}{A}{}\left({t}\right){}{\mathrm{D_x}}{+}{B}{}\left({t}\right){}{\mathrm{D_y}}$ (4.10)
 M > $\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C,X,\mathrm{Gamma},t\right),X\right)$
 $\left\{{\mathrm{_C2}}{}{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}{}{\mathrm{D_x}}{+}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{\mathrm{cos}}{}\left({t}\right)}{}{\mathrm{D_y}}\right\}$ (4.11)

We can use the keyword argument auxiliaryequations to specify an initial position for the vector $X.$

 M > $\mathrm{DGsolve}\left(\mathrm{ParallelTransportEquations}\left(C,X,\mathrm{Gamma},t\right),X,\mathrm{auxiliaryequations}=\left\{A\left(0\right)=1,B\left(0\right)=0\right\}\right)$
 $\left\{{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}{}{\mathrm{D_x}}\right\}$ (4.12)

Example 4.

The source-free Maxwell equations may be expressed in terms of a 2-form $F$ by the equations  and , where $d$ is the exterior derivative and $*$ is the Hodge star operator. In this example we define a 2-form depending on 2 functions of 4 variables and solve the Maxwell equations for $F.$

 M > $\mathrm{DGsetup}\left(\left[x,y,z,t\right],M\right)$
 ${\mathrm{frame name: M}}$ (4.13)
 M > $gâ‰”\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}-\mathrm{dt}&t\mathrm{dt}\right)$
 ${g}{:=}{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{-}{\mathrm{dt}}{}{\mathrm{dt}}$ (4.14)
 M > $Fâ‰”\mathrm{evalDG}\left(A\left(x,y,z,t\right)\mathrm{dx}&w\mathrm{dy}+B\left(x,y,z,t\right)\mathrm{dx}&w\mathrm{dt}\right)$
 ${F}{:=}{A}{}\left({x}{,}{y}{,}{z}{,}{t}\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{B}{}\left({x}{,}{y}{,}{z}{,}{t}\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dt}}$ (4.15)
 M > $\mathrm{DGsolve}\left(\left[\mathrm{ExteriorDerivative}\left(F\right),\mathrm{ExteriorDerivative}\left(\mathrm{HodgeStar}\left(g,F,\mathrm{detmetric}=-1\right)\right)\right],F\right)$
 $\left\{\left({\mathrm{_F1}}{}\left({t}{+}{y}\right){+}{\mathrm{_F2}}{}\left({t}{-}{y}\right)\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}\left({\mathrm{_F1}}{}\left({t}{+}{y}\right){-}{\mathrm{_F2}}{}\left({t}{-}{y}\right){+}{\mathrm{_C1}}\right){}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dt}}\right\}$ (4.16)