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overview of overloaded builtins for LAVF object. Description

 • The functionalities of some Maple builtin commands are extended for use on LAVF object.
 • The following builtins have been overloaded for this purpose: indets, has, type, hastype
 • Let L be a LAVF object.
 • (i) The call type(L, t) returns true if t is any of the following types: module, object, anything, and LAVF. See examples below.
 • (ii) The call type(L, dependent(x)) and type(L, freeof(x)) respectively return true if its determining system S including the DEs system, the independent variables, and the dependent variables of S contain (respectively don't contain) x. See example below.
 • The indets, has, hastype builtin commands accept a LAVF object and apply their methods onto the DEs system, the independent variables, and the dependent variables of the determining system of the LAVF object.
 • These overloaded builtins are 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)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)+\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{η}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{+}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}{,}{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

We first construct a LAVF object for 2-dim Euclidean group 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)

We also construct a fully-integrated LAVF object for E(2).

 > $\mathrm{Lsol}≔\mathrm{LAVFSolve}\left(L,\mathrm{output}="lavf",\mathrm{consts}=\left[a,b,c\right]\right)$
 ${\mathrm{Lsol}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}{a}{}{y}{+}{c}{,}{\mathrm{\eta }}{=}{a}{}{x}{+}{b}\right]\right\}$ (4) type

 > $\mathrm{type}\left(L,'\mathrm{LAVF}'\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{type}\left(L,'\mathrm{object}'\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{type}\left(L,'\mathrm{module}'\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{type}\left(L,\mathrm{dependent}\left(x\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{type}\left(L,\mathrm{freeof}\left(x\right)\right)$
 ${\mathrm{false}}$ (9)

and the fully-integrated LAVF Lsol does contain constant of integration variable a.

 > $\mathrm{type}\left(\mathrm{Lsol},\mathrm{dependent}\left(a\right)\right)$
 ${\mathrm{true}}$ (10) indets, has, hastype

 > $\mathrm{indets}\left(L\right)$
 $\left\{{x}{,}{y}{,}{{\mathrm{\xi }}}_{{y}{,}{y}}{,}{{\mathrm{\eta }}}_{{x}}{,}{{\mathrm{\eta }}}_{{y}}{,}{{\mathrm{\xi }}}_{{x}}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}{,}{\mathrm{\xi }}\right\}$ (11)
 > $\mathrm{indets}\left(\mathrm{Lsol}\right)$
 $\left\{{a}{,}{b}{,}{c}{,}{x}{,}{y}{,}{\mathrm{\eta }}{,}{\mathrm{\xi }}\right\}$ (12)
 > $\mathrm{has}\left(L,b\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{has}\left(\mathrm{Lsol},b\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{hastype}\left(L,\mathrm{scalar}\right)$
 ${\mathrm{true}}$ (15)
 > $\mathrm{hastype}\left(\mathrm{Lsol},\mathrm{float}\right)$
 ${\mathrm{false}}$ (16) Compatibility

 • The LAVF Object Overloaded builtins command was introduced in Maple 2020.