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Copy

clone a LAVF object to have different infinitesimal (and constant of variables) names

 Calling Sequence Copy( obj, vars)

Parameters

 obj - LAVF objects. vars - a name or a list of names

Description

 • Let L be a LAVF object with $n$; infinitesimal variables and $k$; (or maybe none) constants of integration variables. Then the Copy method clones L and returns a new LAVF object with new infinitesimals and constants of integration variable names as given in vars. The new variable names vars will replace the ones from the object.
 • If vars is given as a list of names, then the number of entries in vars must be $n+k$; (i.e. same as the number of dependent variables from the determining system of L) (see GetDependents).
 • If vars is given as one single name $\mathrm{\xi }$, then the new variables names will be ${\mathrm{\xi }}_{1},{\mathrm{\xi }}_{2},\dots ,{\mathrm{\xi }}_{n+k}$.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right),\mathrm{\alpha }\left(x,y\right),\mathrm{\beta }\left(x,y\right),\mathrm{\phi }\left(x,y\right)\right]\right)$
 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

Case 1: A vector fields system for E(2).

 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (4)
 > $\mathrm{Copy}\left(L,\left[\mathrm{\alpha },\mathrm{\beta }\right]\right)$
 $\left[{\mathrm{\alpha }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\beta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\alpha }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\alpha }}}_{{x}}{=}{0}{,}{{\mathrm{\beta }}}_{{x}}{=}{-}{{\mathrm{\alpha }}}_{{y}}{,}{{\mathrm{\beta }}}_{{y}}{=}{0}\right]\right\}$ (5)

Case2 : A vector fields system for E(2) that has been fully integrated.

 > $\mathrm{Lsol}≔\mathrm{LAVFSolve}\left(L,\mathrm{output}="lavf"\right)$
 ${\mathrm{Lsol}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}{\mathrm{_C1}}{}{y}{+}{\mathrm{_C3}}{,}{\mathrm{\eta }}{=}{\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}\right]\right\}$ (6)

Lsol includes infinitesimals ($\mathrm{\xi },\mathrm{\eta }$) and constants of integration _C1, _C2, _C3

 > $\mathrm{DQ}≔\mathrm{GetDeterminingSystem}\left(\mathrm{Lsol}\right)$
 ${\mathrm{DQ}}{≔}\left[{\mathrm{\xi }}{=}{-}{\mathrm{_C1}}{}{y}{+}{\mathrm{_C3}}{,}{\mathrm{\eta }}{=}{\mathrm{_C1}}{}{x}{+}{\mathrm{_C2}}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}{\mathrm{_C1}}{,}{\mathrm{_C2}}{,}{\mathrm{_C3}}\right]$ (7)
 > $\mathrm{Copy}\left(\mathrm{Lsol},\left[\mathrm{\alpha },\mathrm{\beta },a,b,c\right]\right)$
 $\left[{\mathrm{\alpha }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\beta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\alpha }}{=}{-}{a}{}{y}{+}{c}{,}{\mathrm{\beta }}{=}{a}{}{x}{+}{b}\right]\right\}$ (8)
 > $\mathrm{Copy}\left(\mathrm{Lsol},\mathrm{\phi }\right)$
 $\left[{{\mathrm{\phi }}}_{{1}}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{{\mathrm{\phi }}}_{{2}}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\phi }}}_{{1}}{}\left({x}{,}{y}\right){=}{-}{y}{}{{\mathrm{\phi }}}_{{3}}{+}{{\mathrm{\phi }}}_{{5}}{,}{{\mathrm{\phi }}}_{{2}}{}\left({x}{,}{y}\right){=}{x}{}{{\mathrm{\phi }}}_{{3}}{+}{{\mathrm{\phi }}}_{{4}}\right]\right\}$ (9)

Compatibility

 • The Copy command was introduced in Maple 2020.