 Copy - Maple Help

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)
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{ξ}\left(x,y\right),\mathrm{η}\left(x,y\right),\mathrm{α}\left(x,y\right),\mathrm{β}\left(x,y\right),\mathrm{φ}\left(x,y\right)\right]\right)$
 > $V≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (2)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

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

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

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

 > $\mathrm{Lsol}≔\mathrm{LAVFSolve}\left(L,\mathrm{output}="lavf"\right)$
 ${\mathrm{Lsol}}{≔}\left[{\mathrm{\xi }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\xi }}{=}{-}\mathrm{c__1}{}{y}{+}\mathrm{c__3}{,}{\mathrm{\eta }}{=}\mathrm{c__1}{}{x}{+}\mathrm{c__2}\right]\right\}$ (6)

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

 > $\mathrm{DQ}≔\mathrm{GetDeterminingSystem}\left(\mathrm{Lsol}\right)$
 ${\mathrm{DQ}}{≔}\left[{\mathrm{\xi }}{=}{-}\mathrm{c__1}{}{y}{+}\mathrm{c__3}{,}{\mathrm{\eta }}{=}\mathrm{c__1}{}{x}{+}\mathrm{c__2}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{c__3}\right]$ (7)
 > $\mathrm{Copy}\left(\mathrm{Lsol},\left[\mathrm{α},\mathrm{β},a,b,c\right]\right)$
 $\left[{\mathrm{\alpha }}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\beta }}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{\mathrm{\alpha }}{=}{-}{a}{}{y}{+}{c}{,}{\mathrm{\beta }}{=}{a}{}{x}{+}{b}\right]\right\}$ (8)
 > $\mathrm{Copy}\left(\mathrm{Lsol},\mathrm{φ}\right)$
 $\left[{{\mathrm{\phi }}}_{{1}}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{{\mathrm{\phi }}}_{{2}}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\phi }}}_{{1}}{}\left({x}{,}{y}\right){=}{-}{y}{}{{\mathrm{\phi }}}_{{3}}{+}{{\mathrm{\phi }}}_{{5}}{,}{{\mathrm{\phi }}}_{{2}}{}\left({x}{,}{y}\right){=}{x}{}{{\mathrm{\phi }}}_{{3}}{+}{{\mathrm{\phi }}}_{{4}}\right]\right\}$ (9) Compatibility

 • The Copy command was introduced in Maple 2020.