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ImplicitForm

calculate the implicit form of a LAVF object

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ImplicitForm(self)

ImplicitForm(self, infinitesimalsOnly = true)

Parameters

self

-

a LAVF objects.

Description

• 

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 ξa (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  ξa 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.

Examples

with(LieAlgebrasOfVectorFields):

Typesetting:-Settings(userep=true):

Typesetting:-Suppress([xi(x,y),eta(x,y)]):

V := VectorField(xi(x,y)*D[x] + eta(x,y)*D[y], space = [x,y]);

Vξx+ηy

(1)

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]);

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(2)

We first construct a LAVF object for E(2),

L := LAVF(V, E2);

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

And we obtain the fully-integrated LAVF object by solving L,

Ls := LAVFSolve(L, output= "lavf");

Lsξx+ηy&whereξ=_C1y+_C3,η=_C1x+_C2

(4)

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_C1=ξy,_C2=ξyx+η,_C3=ξyy+ξ,ξy,y=0,ξx=0,ηx=ξy,ηy=0,indep=x,y,dep=ξ,η,_C1,_C2,_C3

(5)

Imp is a LHPDE object and has access to various methods.

type(Imp, 'LHPDE');

true

(6)

Ranking of Imp shows that infinitesimals ξ,η are indeed ranked lower than all other variables.

GetRanking(Imp);

_C1,_C2,_C3,ξ,η

(7)

We can also fetch the non-integrated determining system of E2 from Ls, by setting option infinitesimalsOnly = true

S := ImplicitForm(Ls, infinitesimalsOnly = true);

Sξy,y=0,ξx=0,ηx=ξy,ηy=0,indep=x,y,dep=ξ,η

(8)

The non-integrated determining system S should be same as E2

AreSame(S, E2, criteria = "sameSystem");

true

(9)

Compatibility

• 

The ImplicitForm command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LHPDE (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

LAVFSolve

GetRanking

AreSame