detools::arbFuns
-- number of
arbitrary functions in the general solution of an involutive partial
differential equationdetools::arbFuns
(q,alpha)
computes the
number of arbitrary functions in the general solution of an involutive
partial differential equation of order q
and with Cartan
characters alpha
.
detools::arbFuns(q, alpha)
q |
- | the order of the equation: a positive integer. |
alpha |
- | the Cartan characters: a list of nonnegative integers. |
a list of integers of the same length as alpha
. The
i-th entry gives the number of arbitrary functions with
i arguments.
detools::cartan
,
detools::hilbert
detools::arbFuns
performs a purely combinatorial
calculation trying to express the Cartan characters of an involutive
partial differential equations as the number of arbitrary functions in
the general solution. For first order equations this make always sense;
for higher order equations it is possible that negative values occur in
the answer.How many arbitrary functions appear in the general solution of Maxwell's equations in electrodynamics? For three-dimensional space the four Cartan characters are 6, 6, 4 and 0. So we enter
>> detools::arbFuns(1, [6, 6, 4, 0])
[0, 2, 4, 0]
and obtain that (at least formally) the solution space can be parametrised by four functions of three variables and two functions of two variables.