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

 See Also