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(xi(x,y)*D[x] + eta(x,y)*D[y], space = [x,y]);
E2 := LHPDE([diff(xi(x,y),y,y)=0, diff(eta(x,y),x)=-diff(xi(x,y),y), diff(eta(x,y),y)=0, diff(xi(x,y),x)=0], indep = [x,y], dep = [xi, eta]);
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
AreSame(S, E2, criteria = "sameSystem");
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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