InvariantGeometricObjectFields - Maple Help

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GroupActions[InvariantGeometricObjectFields] - find the vector fields, differential forms, tensors or connections which are invariant with respect to a Lie algebra of vector fields

Calling Sequences

InvariantGeometricObjectFields(Gamma, T, options)

Parameters

Gamma     - a list of vector fields on a manifold $M$.

T         - a list of vector fields, differential forms, or tensors on $M$

options   - output = "list", output = "pde", connection = "yes"/"no", coefficientvariables = [x1, x2, ...], unknowns = [F1, F2, ...], ansatz = t , parameters = P

Description

 • Let be a $r$-dimensional Lie algebra of vector fields on a manifold $M$. A vector field, differential form, tensor or connection $S$ is said to be $\mathrm{Γ}$-invariant if the Lie derivative(*) for.
 • The procedure InvariantGeometricObjectFields(Gamma, T) calculates the $\mathrm{Γ}-$invariant geometric object fields which are in the span (over the functions on of the geometric object fields given by the second argument T.
 • The procedure creates the general linear combination of the tensors in T (with coefficients which are functions of the coordinates on and then generates the system of first order PDE for the coefficients arising from the invariance conditions (*).The command pdsolve is used to solve these PDE.
 • If T = [1], then the $\mathrm{Γ}$-invariant functions on are computed.
 • If connection = "yes", then invariant connections are computed.
 • With output = "list", the program returns a basis for the invariant tensors, over the ring of invariant functions. This option is not available when connection = "yes".
 • With output = "pde", the pde system defined by the equations (*) is returned.
 • The exact form for the geometric object fields can be specified by ansatz = t. With this option, the unknown functions in t must be explicitly listed with the unknowns option.
 • If P = {a1, a2, ... } is a set of parameters appearing in Gamma, then the optional argument parameters = P will invoke the case splitting capabilities of pdsolve. Exceptional parameter values will be determined and a sequence of lists of invariant geometry object fields, one list for each set of parameter values, will be returned.
 • Other optional arguments for pdsolve may be passed through the command InvariantGeometricObjectFields.
 • If pdsolve is unable to explicitly solve the pde system defined by LieDerivative(X, t) = 0, then NULL is returned.
 • The command InvariantGeometricObjectFields is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form InvariantGeometricObjectFields(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-InvariantGeometricObjectFields(...).

Examples

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

Define manifolds   with coordinates  and .

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$$\mathrm{DGsetup}\left(\left[x,y\right],N\right):$$\mathrm{DGsetup}\left(\left[x\right],\left[u\right],J,2\right):$

Example 1.

Find all invariant functions, 1-forms, metrics and invariant type [1, 1] tensors for the infinitesimal group of rotations on $M$

 J > $\mathrm{ChangeFrame}\left(M\right)$
 ${J}$ (2.1)
 M > $\mathrm{Γ1}≔\mathrm{evalDG}\left(\left[x\mathrm{D_y}-y\mathrm{D_x},x\mathrm{D_z}-z\mathrm{D_x},y\mathrm{D_z}-z\mathrm{D_y}\right]\right)$
 ${\mathrm{Γ1}}{:=}\left[{-}{\mathrm{D_x}}{}{y}{+}{\mathrm{D_y}}{}{x}{,}{-}{\mathrm{D_x}}{}{z}{+}{\mathrm{D_z}}{}{x}{,}{-}{\mathrm{D_y}}{}{z}{+}{\mathrm{D_z}}{}{y}\right]$ (2.2)

Invariant Functions:

 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},\left[1\right]\right)$
 ${\mathrm{_F1}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)$ (2.3)
 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},\left[1\right],\mathrm{output}="list"\right)$
 $\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right]$ (2.4)

Invariant 1-forms:

 M > $T≔\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right]$
 ${T}{:=}\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{\mathrm{dz}}\right]$ (2.5)
 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},T,\mathrm{output}="list"\right)$
 $\left[\frac{\sqrt{{-}{{z}}^{{2}}}{}{x}{}{\mathrm{dx}}}{{z}}{+}\frac{\sqrt{{-}{{z}}^{{2}}}{}{y}{}{\mathrm{dy}}}{{z}}{+}\sqrt{{-}{{z}}^{{2}}}{}{\mathrm{dz}}\right]$ (2.6)

Note that the format of the answer can be improved with the assuming command.

 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},T,\mathrm{output}="list"\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}0
 $\left[\frac{{x}{}{\mathrm{dx}}}{{z}}{+}\frac{{y}{}{\mathrm{dy}}}{{z}}{+}{\mathrm{dz}}\right]$ (2.7)

Invariant Metrics:

 M > $T≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],2\right)$
 ${T}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.8)
 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},T,\mathrm{output}="list"\right)$
 $\left[{\mathrm{dx}}{}{\mathrm{dx}}{+}{\mathrm{dy}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{dz}}{,}{x}{}{z}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{{x}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{x}{}{y}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{x}{}{y}{}{\mathrm{dy}}{}{\mathrm{dx}}{+}{{y}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{z}{}{y}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{x}{}{z}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}{z}{}{y}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}{{z}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.9)

Invariant [1, 1] Tensors:

 M > $T≔\mathrm{GenerateTensors}\left(\left[\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]\right]\right)$
 ${T}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}\right]$ (2.10)
 M > $\mathrm{_EnvExplicit}≔\mathrm{true}$
 ${\mathrm{_EnvExplicit}}{:=}{\mathrm{true}}$ (2.11)
 M > $\mathrm{Inv}≔\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ1},T,\mathrm{output}="list"\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}0
 ${\mathrm{Inv}}{:=}\left[{\mathrm{D_x}}{}{\mathrm{dx}}{+}{\mathrm{D_y}}{}{\mathrm{dy}}{+}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{-}{\mathrm{D_x}}{}{\mathrm{dx}}{}{z}{-}{\mathrm{D_y}}{}{\mathrm{dy}}{}{z}{+}{\mathrm{D_z}}{}{\mathrm{dx}}{}{x}{+}{\mathrm{D_z}}{}{\mathrm{dy}}{}{y}{,}{-}{\mathrm{D_x}}{}{\mathrm{dy}}{}{z}{+}{\mathrm{D_y}}{}{\mathrm{dx}}{}{z}{-}{\mathrm{D_z}}{}{\mathrm{dx}}{}{y}{+}{\mathrm{D_z}}{}{\mathrm{dy}}{}{x}\right]$ (2.12)

Example 2.

Find the vector fields which commute with the Lie algebra of vector fields ${\mathrm{Γ}}_{2}$.

 M > $\mathrm{ChangeFrame}\left(M\right):$
 M > $\mathrm{Γ2}≔\left[\mathrm{D_x},\mathrm{exp}\left(-y\right)\mathrm{D_z},x\mathrm{D_x}+\mathrm{D_y}\right]$
 ${\mathrm{Γ2}}{:=}\left[{\mathrm{D_x}}{,}{{ⅇ}}^{{-}{y}}{}{\mathrm{D_z}}{,}{\mathrm{D_x}}{}{x}{+}{\mathrm{D_y}}\right]$ (2.13)
 M > $T≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${T}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.14)
 M > $Z≔\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ2},T,\mathrm{output}="list"\right)$
 ${Z}{:=}\left[{\mathrm{D_z}}{}{z}{-}{\mathrm{D_y}}{,}{\mathrm{D_z}}{,}{{ⅇ}}^{{y}}{}{\mathrm{D_x}}\right]$ (2.15)

Give the partial differential equations which were solved to calculate the commuting vectors in the list Z.

 J > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ2},T,\mathrm{unknowns}=\left[A\left(x,y,z\right),B\left(x,y,z\right),C\left(x,y,z\right)\right],\mathrm{output}="pde"\right)$
 $\left[\frac{{\partial }}{{\partial }{x}}{}{A}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{x}}{}{B}{}\left({x}{,}{y}{,}{z}\right){,}\frac{{\partial }}{{\partial }{x}}{}{C}{}\left({x}{,}{y}{,}{z}\right){,}{{ⅇ}}^{{-}{y}}{}\left(\frac{{\partial }}{{\partial }{z}}{}{A}{}\left({x}{,}{y}{,}{z}\right)\right){,}{{ⅇ}}^{{-}{y}}{}\left(\frac{{\partial }}{{\partial }{z}}{}{B}{}\left({x}{,}{y}{,}{z}\right)\right){,}{{ⅇ}}^{{-}{y}}{}\left({B}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{z}}{}{C}{}\left({x}{,}{y}{,}{z}\right)\right){,}{-}{A}{}\left({x}{,}{y}{,}{z}\right){+}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{A}{}\left({x}{,}{y}{,}{z}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{A}{}\left({x}{,}{y}{,}{z}\right){,}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{B}{}\left({x}{,}{y}{,}{z}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{B}{}\left({x}{,}{y}{,}{z}\right){,}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{C}{}\left({x}{,}{y}{,}{z}\right)\right){+}\frac{{\partial }}{{\partial }{y}}{}{C}{}\left({x}{,}{y}{,}{z}\right){,}{0}\right]{,}\left[{A}{}\left({x}{,}{y}{,}{z}\right){,}{B}{}\left({x}{,}{y}{,}{z}\right){,}{C}{}\left({x}{,}{y}{,}{z}\right)\right]$ (2.16)

Find the vector fields of the special form  + c(x)D_z which commute with ${\mathrm{Γ}}_{2}$.

 M > $Z≔\mathrm{evalDG}\left(a\left(x\right)\mathrm{D_x}+b\left(x,y\right)\mathrm{D_y}+c\left(x,y,z\right)\mathrm{D_z}\right)$
 ${Z}{:=}{a}{}\left({x}\right){}{\mathrm{D_x}}{+}{b}{}\left({x}{,}{y}\right){}{\mathrm{D_y}}{+}{c}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{D_z}}$ (2.17)
 M > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ2},Z,\mathrm{unknowns}=\left[a\left(x\right),b\left(x,y\right),c\left(x,y,z\right)\right],\mathrm{output}="list"\right)$
 $\left[{\mathrm{D_z}}{}{z}{-}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.18)

Example 3.

Find the second and third order differential invariants for the infinitesimal Euclidean group acting on the plane.

 M > $\mathrm{ChangeFrame}\left(J\right):$
 J > $\mathrm{Γ3}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_u}\left[\right],u\left[\right]\mathrm{D_x}-x\mathrm{D_u}\left[\right]\right]\right)$
 ${\mathrm{Γ3}}{:=}\left[{\mathrm{D_x}}{,}{{\mathrm{D_u}}}_{{[}{]}}{,}{\mathrm{D_x}}{}{{u}}_{{[}{]}}{-}{x}{}{{\mathrm{D_u}}}_{{[}{]}}\right]$ (2.19)
 J > $\mathrm{Gamma3a}≔\mathrm{map}\left(\mathrm{Prolong},\mathrm{Γ3},2\right)$
 ${\mathrm{Gamma3a}}{:=}\left[{\mathrm{D_x}}{,}{{\mathrm{D_u}}}_{{[}{]}}{,}{{u}}_{{[}{]}}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{{[}{]}}{-}\left({{u}}_{{1}}^{{2}}{+}{1}\right){}{{\mathrm{D_u}}}_{{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}\right]$ (2.20)
 J > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma3a},\left[1\right],\mathrm{coefficientvariables}=\left[x,u\left[\right],u\left[1\right],u\left[1,1\right]\right]\right)$
 ${\mathrm{_F1}}{}\left(\frac{{{u}}_{{1}{,}{1}}}{{\left({{u}}_{{1}}^{{2}}{+}{1}\right)}^{{3}{/}{2}}}\right)$ (2.21)
 J > $\mathrm{Gamma2b}≔\mathrm{map}\left(\mathrm{Prolong},\mathrm{Gamma3a},3\right)$
 ${\mathrm{Gamma2b}}{:=}\left[{\mathrm{D_x}}{,}{{\mathrm{D_u}}}_{{[}{]}}{,}{{u}}_{{[}{]}}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{{[}{]}}{-}\left({{u}}_{{1}}^{{2}}{+}{1}\right){}{{\mathrm{D_u}}}_{{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}{-}\left({4}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{3}{}{{u}}_{{1}{,}{1}}^{{2}}\right){}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{1}}\right]$ (2.22)
 J > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b},\left[1\right]\right)$
 ${\mathrm{_F1}}{}\left(\frac{{{u}}_{{1}{,}{1}}}{{\left({{u}}_{{1}}^{{2}}{+}{1}\right)}^{{3}{/}{2}}}{,}\frac{{{u}}_{{1}}^{{2}}{}{{u}}_{{1}{,}{1}{,}{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}^{{2}}{+}{{u}}_{{1}{,}{1}{,}{1}}}{{\left({{u}}_{{1}}^{{2}}{+}{1}\right)}^{{3}}}\right)$ (2.23)

Find the invariant Lagrangians on the 1-jet.

 J > $S≔\mathrm{Dx}$
 ${S}{:=}{\mathrm{Dx}}$ (2.24)
 J > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b},\left[S\right],\mathrm{coefficientvariables}=\left[x,u\left[\right],u\left[1\right]\right]\right)$
 ${\mathrm{_C1}}{}\sqrt{{{u}}_{{1}}^{{2}}{+}{1}}{}{\mathrm{Dx}}$ (2.25)

Find the invariant "source" forms on the 2-jet.

 J > $S≔\mathrm{Dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Cu}\left[\right]$
 ${S}{:=}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{[}{]}}$ (2.26)
 J > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Gamma2b},\left[S\right],\mathrm{coefficientvariables}=\left[x,u\left[\right],u\left[1\right],u\left[1,1\right]\right]\right)$
 ${\mathrm{_F1}}{}\left(\frac{{{u}}_{{1}{,}{1}}}{{\left({{u}}_{{1}}^{{2}}{+}{1}\right)}^{{3}{/}{2}}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{[}{]}}$ (2.27)

Example 4.

Find the invariant 1-forms for a list of vector fields depending on a parameter alpha.

 J > $\mathrm{ChangeFrame}\left(N\right)$
 ${J}$ (2.28)
 N > $\mathrm{Γ4}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},x\mathrm{D_x}+\mathrm{\alpha }y\mathrm{D_y}\right]\right)$
 ${\mathrm{Γ4}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{}{y}{}{\mathrm{α}}{+}{\mathrm{D_x}}{}{x}\right]$ (2.29)
 N > $F≔\left[\mathrm{dx},\mathrm{dy}\right]$
 ${F}{:=}\left[{\mathrm{dx}}{,}{\mathrm{dy}}\right]$ (2.30)
 N > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ4},F,\mathrm{parameters}=\left\{\mathrm{\alpha }\right\}\right)$
 ${\mathrm{_F1}}{}\left({y}\right){}{\mathrm{dy}}{,}{\mathrm{_C2}}{}{{y}}^{{-}\frac{{1}}{{\mathrm{α}}}}{}{\mathrm{dx}}{+}\frac{{\mathrm{_C1}}{}{\mathrm{dy}}}{{y}}{,}\left[\left\{{\mathrm{α}}{=}{0}\right\}{,}\left\{{\mathrm{α}}{=}{\mathrm{α}}\right\}\right]$ (2.31)
 N > $\mathrm{InvariantGeometricObjectFields}\left(\mathrm{Γ4},F,\mathrm{output}="list",\mathrm{parameters}=\left\{\mathrm{\alpha }\right\}\right)$
 $\left[{\mathrm{dy}}\right]{,}\left[\frac{{\mathrm{dy}}}{{y}}{,}{{y}}^{{-}\frac{{1}}{{\mathrm{α}}}}{}{\mathrm{dx}}\right]{,}\left[\left\{{\mathrm{α}}{=}{0}\right\}{,}\left\{{\mathrm{α}}{=}{\mathrm{α}}\right\}\right]$ (2.32)

Example 5.

The command InvariantGeometricObjectFields can also be used to calculate tensors on a Lie algebra which are invariant with respect to a subalgebra.

Retrieve a Lie algebra from the DifferentialGeometry library.

 > $\mathrm{LD}≔\mathrm{Library}:-\mathrm{Retrieve}\left("Winternitz",1,\left[4,10\right],\mathrm{alg1}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.33)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg1}}$ (2.34)
 alg1 > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{θ1},\mathrm{θ2},\mathrm{θ3},\mathrm{θ4}\right],2\right)$
 ${S}{:=}\left[{\mathrm{θ1}}{}{\mathrm{θ1}}{,}\frac{{1}}{{2}}{}{\mathrm{θ1}}{}{\mathrm{θ2}}{+}\frac{{1}}{{2}}{}{\mathrm{θ2}}{}{\mathrm{θ1}}{,}\frac{{1}}{{2}}{}{\mathrm{θ1}}{}{\mathrm{θ3}}{+}\frac{{1}}{{2}}{}{\mathrm{θ3}}{}{\mathrm{θ1}}{,}\frac{{1}}{{2}}{}{\mathrm{θ1}}{}{\mathrm{θ4}}{+}\frac{{1}}{{2}}{}{\mathrm{θ4}}{}{\mathrm{θ1}}{,}{\mathrm{θ2}}{}{\mathrm{θ2}}{,}\frac{{1}}{{2}}{}{\mathrm{θ2}}{}{\mathrm{θ3}}{+}\frac{{1}}{{2}}{}{\mathrm{θ3}}{}{\mathrm{θ2}}{,}\frac{{1}}{{2}}{}{\mathrm{θ2}}{}{\mathrm{θ4}}{+}\frac{{1}}{{2}}{}{\mathrm{θ4}}{}{\mathrm{θ2}}{,}{\mathrm{θ3}}{}{\mathrm{θ3}}{,}\frac{{1}}{{2}}{}{\mathrm{θ3}}{}{\mathrm{θ4}}{+}\frac{{1}}{{2}}{}{\mathrm{θ4}}{}{\mathrm{θ3}}{,}{\mathrm{θ4}}{}{\mathrm{θ4}}\right]$ (2.35)

Find the symmetric rank 2 tensors on alg1 which are invariant with respect to the subalgebra spanned by

 alg1 > $\mathrm{InvariantGeometricObjectFields}\left(\left[\mathrm{e1},\mathrm{e2}\right],S,\mathrm{output}="list"\right)$
 $\left[\frac{{1}}{{2}}{}{\mathrm{θ1}}{}{\mathrm{θ4}}{+}\frac{{1}}{{2}}{}{\mathrm{θ3}}{}{\mathrm{θ3}}{+}\frac{{1}}{{2}}{}{\mathrm{θ4}}{}{\mathrm{θ1}}{,}{\mathrm{θ2}}{}{\mathrm{θ2}}{,}\frac{{1}}{{2}}{}{\mathrm{θ2}}{}{\mathrm{θ4}}{+}\frac{{1}}{{2}}{}{\mathrm{θ4}}{}{\mathrm{θ2}}{,}{\mathrm{θ4}}{}{\mathrm{θ4}}\right]$ (2.36)