Eta_k(S, DepVars, 'options'='value')

Consider the generic form of a list of infinitesimals of a PDE problem in - say - two independent and two dependent variables u(x, t), v(x, t): there are then two  infinitesimals associated to each the independent variables and two  infinitesimals associated to each of the dependent variables, as in

For illustration purposes consider also the infinitesimal generator operator corresponding to this list of infinitesimals

The table-procedure returned by Eta_k, able to compute any prolongation, is constructed instantly and without consuming any computational resources, equally from  or

This table output is always displayed as . The prolongation of order "zero" of the first dependent infinitesimal, that is, the  infinitesimal associated to the dependent variable  is obtained by entering  or  and is just equal to  respectively written in jetvariables or jetnumbers notation

The difference between jetvariables and jetnumbers is more evident when actually computing prolongations, for example for the first prolongation of  with respect to

In this output (4.10) expressed using jetnumbers  represents ,  represents the derivative with respect to ,  the second derivative with respect to  and so on; the convention is the same one used by the Maple D differentiation operator.

On the other hand, in the output (4.9) expressed in jetvariables,  represents ,  represents the derivative with respect to ,  the second derivative with respect to  and so on; this convention is also used in all the symmetry commands of PDEtools, the declare command in the mathematical display of derivatives, and by the DifferentialAlgebra package.

To avoid redundant display in the following output use the declare facility, which will also make the derivatives be displayed compactly, indexed

For example, the now compact display for  lines above is

There are three possible prolongations of order two for each of the  and  infinitesimals, these are: two times with respect to the first independent variable, ditto with respect to the second independent variable, and the mixed prolongation; for  these are

Note the compact display in the output above, due to the use of declare. To see the contents behind this compact display use show

It is sometimes possible to compactify these expressions a bit more by using, for instance, simplify,size or convert,horner.

You can also request a particular prolongation directly, either using jetvariables or jetnumbers notation, with or without the option expanded

Re-expressing the output above in jetvariables notation instead of jetnumbers we get the same as from  in (4.15)

The Eta_k command also works with anticommutative variables, natively, without using the approach explained in PerformOnAnticommutativeSystem.

Set first  and  as suffixes for variables of type/anticommutative (see Setup)

A PDE system example with one unknown anticommutative function  of four variables, two commutative and two anticommutative; to avoid redundant typing in the input that follows and redundant display of information on the screen, use PDEtools:-declare, and PDEtools:-diff_table, that also handles anticommutative variables by automatically using Physics:-diff when Physics is loaded

Now we can enter derivatives directly as the function's name indexed by the differentiation variables and see the display the same way; two PDEs

Set now the generic form of the infinitesimals for a PDE system like this one formed by pde[1] and pde[2]. For this purpose, we need anticommutative infinitesimals for the dependent variable  and two of the independent variables,  and ; we use here the capital greek letters  and  for the anticommutative infinitesimal symmetry generators and  for the commutative ones

The corresponding InfinitesimalGenerator

The table-function that returns the prolongation of the infinitesimal for  is computed with Eta_k, assign it here to the lower case  to use more familiar notation

The first prolongations of  with respect to  and  

The second mixed prolongations of  with respect to  and  

convert,horner, D, declare, Infinitesimals, PDEtools, Physics, Setup, simplify,size, ToJet