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liesymm

 Eta
 Compute the coefficients of the generator of a finite point transformation

 Calling Sequence Eta(f, x) Eta[1](f, x) Eta[2](f, x, y) Eta[3](f, x, y, z)

Parameters

 f - named partial in the sense of depvars() x, y, z - Independent variable in the sense of indepvars()

Description

 • This is a special differential operator defined in terms of TD. The result is an inert expression reported in terms of Diff procedure.  The result can be forced to evaluate further by use of dvalue() or value(), but any variable dependencies for unknown functions must be defined prior to such evaluation. Such variable dependencies can be explicitly specified by use of vfix().
 • It arises in the course of extending the generator for the finite point transformations to the partial derivatives and is in fact computes the coefficient of the various partials in that generator.
 • This routine is part of the liesymm package and is ordinarily loaded via with(liesymm). It can also be called via the package style'' name liesymm[Eta].

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$
 > $\mathrm{indepvars}\left(x,y\right)$
 $\left[{x}{,}{y}\right]$ (1)
 > $\mathrm{depvars}\left(f,g\right)$
 $\left[{f}{,}{g}\right]$ (2)
 > $\mathrm{Η}\left(f,x\right)$
 $\frac{{ⅆ}{\mathrm{V3}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V3}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V3}}}{{ⅆ}{g}}\right){-}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w2}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)$ (3)
 > $\mathrm{w1}=\mathrm{Diff}\left(\mathrm{translate}\left(\mathrm{w1}\right)\right)$
 ${\mathrm{w1}}{=}\frac{{ⅆ}{f}}{{ⅆ}{x}}$ (4)
 > ${\mathrm{Η}}_{2}\left(g,x,y\right)$
 $\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right){+}{\mathrm{w2}}{}\left(\frac{{\partial }}{{\partial }{f}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){+}{\mathrm{w4}}{}\left(\frac{{\partial }}{{\partial }{g}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){+}{\mathrm{w6}}{}\left(\frac{{\partial }}{{\partial }{\mathrm{w1}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){+}{\mathrm{w8}}{}\left(\frac{{\partial }}{{\partial }{\mathrm{w2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){+}{\mathrm{w10}}{}\left(\frac{{\partial }}{{\partial }{\mathrm{w3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){+}{\mathrm{w12}}{}\left(\frac{{\partial }}{{\partial }{\mathrm{w4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V4}}}{{ⅆ}{g}}\right){-}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right)\right){-}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{x}}{+}{\mathrm{w1}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w3}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right)\right)\right)\right){-}{\mathrm{w9}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{y}}{+}{\mathrm{w2}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{f}}\right){+}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{g}}\right){+}{\mathrm{w6}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{\mathrm{w1}}}\right){+}{\mathrm{w8}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{\mathrm{w2}}}\right){+}{\mathrm{w10}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{\mathrm{w3}}}\right){+}{\mathrm{w12}}{}\left(\frac{{ⅆ}{\mathrm{V1}}}{{ⅆ}{\mathrm{w4}}}\right)\right){-}{\mathrm{w10}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{y}}{+}{\mathrm{w2}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{f}}\right){+}{\mathrm{w4}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{g}}\right){+}{\mathrm{w6}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{\mathrm{w1}}}\right){+}{\mathrm{w8}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{\mathrm{w2}}}\right){+}{\mathrm{w10}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{\mathrm{w3}}}\right){+}{\mathrm{w12}}{}\left(\frac{{ⅆ}{\mathrm{V2}}}{{ⅆ}{\mathrm{w4}}}\right)\right)$ (5)