ImplicitForm - Maple Programming Help

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ImplicitForm

calculate the implicit form of a LAVF object

 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 $\mathrm{\xi }\ll a$ (i.e. all $\mathrm{\xi }$'s are ranked lower than any of $a$'s) where $\mathrm{\xi }=\left({\mathrm{\xi }}^{1},,{\mathrm{\xi }}^{n}\right)$ are infinitesimals and $a=\left({a}^{1},..,{a}^{t}\right)$are non-infinitesimals such as constants of integration variables.
 • The returned output, a LHPDE object, is in rif-reduced form with ranking  $\mathrm{\xi }\ll a$ 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.

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]);
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (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]);
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

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

 > L := LAVF(V, E2);
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)

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

 > Ls := LAVFSolve(L, output= "lavf");
 ${\mathrm{Ls}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}{\mathrm{_C1}}{}{y}{+}{\mathrm{_C3}}{,}{\mathrm{\eta }}{=}{\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}\right]\right\}$ (4)

As we can see Ls has infinitesimals $\mathrm{\xi },\mathrm{\eta }$ and constant of integration variables _C1, _C2, _C3.  Now let's find the implicit form of Ls,

 > Imp := ImplicitForm(Ls);
 ${\mathrm{Imp}}{≔}\left[{\mathrm{_C1}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{_C2}}{=}\left({{\mathrm{\xi }}}_{{y}}\right){}{x}{+}{\mathrm{\eta }}{,}{\mathrm{_C3}}{=}{-}\left({{\mathrm{\xi }}}_{{y}}\right){}{y}{+}{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}{\mathrm{_C1}}{,}{\mathrm{_C2}}{,}{\mathrm{_C3}}\right]$ (5)

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

 > type(Imp, 'LHPDE');
 ${\mathrm{true}}$ (6)

Ranking of Imp shows that infinitesimals $\mathrm{\xi },\mathrm{\eta }$ are indeed ranked lower than all other variables.

 > GetRanking(Imp);
 $\left[\left[{\mathrm{_C1}}{,}{\mathrm{_C2}}{,}{\mathrm{_C3}}\right]{,}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]\right]$ (7)

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

 > S := ImplicitForm(Ls, infinitesimalsOnly = true);
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (8)

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

 > AreSame(S, E2, criteria = "sameSystem");
 ${\mathrm{true}}$ (9)

Compatibility

 • The ImplicitForm command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.