calculate the implicit form of a LAVF object
ImplicitForm(self, infinitesimalsOnly = true)
a LAVF objects.
Let L be a LAVF object that is either partial or fully integrated (i.e. its determining system includes constants or functions that are not infinitesimals). Then ImplicitForm(L) returns the implicit form of L, as a new LHPDE object.
The implicit form of L is defined by rif-reducing its determining system with respect to a block ranking ξ≪a (i.e. all ξ's are ranked lower than any of a's) where ξ=ξ1,,ξn are infinitesimals and a=a1,..,atare non-infinitesimals such as constants of integration variables.
The returned output, a LHPDE object, is in rif-reduced form with ranking ξ≪a recorded. See Overview of the LHPDE object for more detail.
In the second calling sequence, the call returns a sub-system that includes infinitesimals only from the implicit form of L. This 'infinitesimals-only' sub-system is same as the non-integrated determining system of L.
If the input LAVF object is non-integrated (i.e. no constants of integration variables), then the implicit form of L is its determining system itself.
This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.
V ≔ VectorField⁡ξ⁡x,y⁢Dx+η⁡x,y⁢Dy,space=x,y
E2 ≔ LHPDE⁡∂2∂y2⁢ξ⁡x,y=0,∂∂x⁢η⁡x,y=−∂∂y⁢ξ⁡x,y,∂∂y⁢η⁡x,y=0,∂∂x⁢ξ⁡x,y=0,indep=x,y,dep=ξ,η
We first construct a LAVF object for E(2),
L ≔ LAVF⁡V,E2
And we obtain the fully-integrated LAVF object by solving L,
Ls ≔ LAVFSolve⁡L,output=lavf
As we can see Ls has infinitesimals ξ,η and constant of integration variables _C1, _C2, _C3. Now let's find the implicit form of Ls,
Imp ≔ ImplicitForm⁡Ls
Imp is a LHPDE object and has access to various methods.
Ranking of Imp shows that infinitesimals ξ,η are indeed ranked lower than all other variables.
We can also fetch the non-integrated determining system of E2 from Ls, by setting option infinitesimalsOnly = true
S ≔ ImplicitForm⁡Ls,infinitesimalsOnly=true
The non-integrated determining system S should be same as E2
The ImplicitForm command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
LieAlgebrasOfVectorFields (Package overview)
LAVF (Object overview)
LHPDE (Object overview)
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