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detools::arbFuns -- number of arbitrary functions in the general solution of an involutive partial differential equation

Introduction

detools::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.

Call(s)

detools::arbFuns(q, alpha)

Parameters

q - the order of the equation: a positive integer.
alpha - the Cartan characters: a list of nonnegative integers.

Returns

a list of integers of the same length as alpha. The i-th entry gives the number of arbitrary functions with i arguments.

Related Functions

detools::cartan, detools::hilbert

Details

Example 1

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.

Background




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