 ImplicitForm - Maple Help

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

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

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

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

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

 > $\mathrm{Ls}≔\mathrm{LAVFSolve}\left(L,\mathrm{output}="lavf"\right)$
 ${\mathrm{Ls}}{≔}\left[{\mathrm{\xi }}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}\mathrm{c__1}{y}{+}\mathrm{c__3}{,}{\mathrm{\eta }}{=}\mathrm{c__1}{x}{+}\mathrm{c__2}\right]\right\}$ (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,

 > $\mathrm{Imp}≔\mathrm{ImplicitForm}\left(\mathrm{Ls}\right)$
 ${\mathrm{Imp}}{≔}\left[\mathrm{c__1}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}\mathrm{c__2}{=}\left({{\mathrm{\xi }}}_{{y}}\right){x}{+}{\mathrm{\eta }}{,}\mathrm{c__3}{=}{-}\left({{\mathrm{\xi }}}_{{y}}\right){y}{+}{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{c__3}\right]$ (5)

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

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

 > $\mathrm{GetRanking}\left(\mathrm{Imp}\right)$
 $\left[\left[\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{c__3}\right]{,}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]\right]$ (7)

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

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

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

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

 • The ImplicitForm command was introduced in Maple 2020.