liesymm

extended differential operator


Calling Sequence


TD(F, x)
TD[1](F, x)
TD[i](F, x)


Parameters


F



algebraic expression

x



independent variable

i



integer indicating the level of extension





Description


•

This routine constructs the derivative of the expression F with respect to the independent variable x. It has been extended in the sense that the expression F is regarded as a function of the dependent variables (as reported by depvars()) and the independent variables and various named partial derivatives nested to higher and higher levels depending on the index. The chain rule is applied to reflect these dependencies.

•

The second argument is restricted to be one of the independent variables as specified by indepvars().

•

A subscripted name such as ${\mathrm{TD}}_{2}$ indicates that a certain ``level of extension'' is to be used. For example, an extension level of 2 indicates that F is to be regarded as a function of the dependent variables and all their first partials with respect to indepvars().

•

If names representing the required partials have not already been created (for example, through the use of makeform() or determine()) then variables of the form wi are automatically introduced as names for the various partial derivatives involved, as determined by the level of extension requested.

•

The default extension level is 1 so that TD is equivalent to ${\mathrm{TD}}_{1}$.

•

The independent and dependent variables must have been specified prior to use of TD. This can be done directly through use of the commands indepvars() and depvars(), or indirectly through the use of makeforms().

•

The mapping of names such as w1 to partial derivatives can be obtained through use of the command translate().

•

The result is an inert expression reported in terms of Diff procedure. It 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().

•

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[TD].



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) 
>

$\mathrm{TD}\left(F\,x\right)$

$\frac{{\partial}}{{\partial}{x}}{}{F}{\+}{\mathrm{w1}}{}\left(\frac{{\partial}}{{\partial}{f}}{}{F}\right){\+}{\mathrm{w3}}{}\left(\frac{{\partial}}{{\partial}{g}}{}{F}\right)$
 (3) 
>

$\mathrm{w1}\=\mathrm{Diff}\left(\mathrm{translate}\left(\mathrm{w1}\right)\right)$

${\mathrm{w1}}{\=}\frac{{\partial}}{{\partial}{x}}{}{f}$
 (4) 
>

$\mathrm{TD}\[2\]\left(F\,y\right)$

$\frac{{\partial}}{{\partial}{y}}{}{F}{\+}{\mathrm{w2}}{}\left(\frac{{\partial}}{{\partial}{f}}{}{F}\right){\+}{\mathrm{w4}}{}\left(\frac{{\partial}}{{\partial}{g}}{}{F}\right){\+}{\mathrm{w6}}{}\left(\frac{{\partial}}{{\partial}{\mathrm{w1}}}{}{F}\right){\+}{\mathrm{w8}}{}\left(\frac{{\partial}}{{\partial}{\mathrm{w2}}}{}{F}\right){\+}{\mathrm{w10}}{}\left(\frac{{\partial}}{{\partial}{\mathrm{w3}}}{}{F}\right){\+}{\mathrm{w12}}{}\left(\frac{{\partial}}{{\partial}{\mathrm{w4}}}{}{F}\right)$
 (5) 






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