DiffAlg -- differential algebra

Description

DiffAlg is a differential algebra package. It can compute the usual operations with polynomial differential forms and vector fields. Its main purpose is to associate algebraic objects to differential operators in the exterior algebra of differential forms.

The simplest way to load the package is with the command:

loadPackage "DiffAlg"

Then, one can define a linear differential 1-form, w, and the radial vector field, R, in 3-dimensional space as:

i1 : w = newForm(2,1,1,"a")

o1 = (a x  + a x  + a x )dx  + (a x  + a x  + a x )dx  + (a x  + a x  +
       0 0    3 1    6 2   0     1 0    4 1    7 2   1     2 0    5 1  
     ------------------------------------------------------------------------
     a x )dx
      8 2   2

o1 : DiffAlgForm

i2 : R = radial 2

o2 = x ax  + x ax  + x ax
      0  0    1  1    2  2

o2 : DiffAlgField

i3 : ring w

      QQ[i]
o3 = ------[][a ..a ][x ..x ][dx ..dx ]
      2        0   8   0   2    0    2
     i  + 1

o3 : PolynomialRing, 3 skew commutative variable(s)

i4 : ring R

      QQ[i]
o4 = ------[][x ..x ][ax ..ax ]
      2        0   2    0    2
     i  + 1

o4 : PolynomialRing

All possible options to call the package can be given with the command:

loadPackage ("DiffAlg",Configuration => {"BaseRing" => aRing, "VariableName" => varSymbol, "DiffName" => difSymbol, "FieldName" => derSymbol})

where:

aRing, a Ring, the base ring. Default QQ[i]
varSymbol, a String, the name of the affine coordinates. Default x
difSymbol, a String, the symbol to denote the differential of a coordinate. Default d
derSymbol, a String, the symbol to denote the partial derivative of a coordinate. Default a

Caveat

It is recommended to operate in low degrees and dimensions because of the computational time needed to handle the number of variables generated in every degree.

Authors

Certification

Version 1.5 of this package was accepted for publication in volume 9 of The Journal of Software for Algebra and Geometry on 19 November 2018, in the article DiffAlg: a Differential algebra package. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.5 of DiffAlg.

Source code

The source code from which this documentation is derived is in the file DiffAlg.m2.

Exports

Types
- DiffAlgDistribution -- the class of distributions of vector fields
- DiffAlgElement -- the class of all differential forms and vector fields
- DiffAlgField -- the class of all vector fields
- DiffAlgForm -- the class of all differential forms
Functions and commands
- dist -- produces a DiffAlgDistribution from a list
- genIm -- a basis of the image of a linear expression
- genKer -- basis of the kernel of a linear expression
- isInvolutive -- tests if a distribution is involutive
- linearComb -- generic linear combination of elements
- logarithmicForm -- creates a logarithmic form
- moduliIdeal -- ideal generated by the coefficients of a differential form or vector field
- newField -- constructor of a vector field
- newForm -- constructor of a differential form
- projectivize -- projectivize a differential form or vector field
- radial -- defines the radial vector field
- singularIdeal -- ideal generated by the polynomial coefficients of a differential form or vector field
Methods
- - DiffAlgElement -- negation of a differential form or vector field
- degree(DiffAlgElement) -- degree of a differential form or a vector field
- diff(DiffAlgForm) -- exterior differential
- DiffAlgElement * QQ -- see DiffAlgElement * RingElement -- scalar multiplication
- DiffAlgElement * RingElement -- scalar multiplication
- DiffAlgElement * ZZ -- see DiffAlgElement * RingElement -- scalar multiplication
- QQ * DiffAlgElement -- see DiffAlgElement * RingElement -- scalar multiplication
- RingElement * DiffAlgElement -- see DiffAlgElement * RingElement -- scalar multiplication
- ZZ * DiffAlgElement -- see DiffAlgElement * RingElement -- scalar multiplication
- DiffAlgElement + DiffAlgElement -- addition
- DiffAlgElement - DiffAlgElement -- subtraction
- DiffAlgElement / QQ -- see DiffAlgElement / RingElement -- scalar division
- DiffAlgElement / RingElement -- scalar division
- DiffAlgElement / ZZ -- see DiffAlgElement / RingElement -- scalar division
- DiffAlgElement | String -- concatenate a string with a differential form or vector field
- String | DiffAlgElement -- see DiffAlgElement | String -- concatenate a string with a differential form or vector field
- DiffAlgField _ DiffAlgForm -- contraction of a differential form with respect to a vector field
- DiffAlgForm _ DiffAlgField -- see DiffAlgField _ DiffAlgForm -- contraction of a differential form with respect to a vector field
- DiffAlgField | DiffAlgField -- Lie bracket
- DiffAlgForm * DiffAlgForm -- see DiffAlgForm ^ DiffAlgForm -- exterior product
- DiffAlgForm ^ DiffAlgForm -- exterior product
- dist(List) -- see dist -- produces a DiffAlgDistribution from a list
- genIm(DiffAlgElement,DiffAlgElement) -- see genIm -- a basis of the image of a linear expression
- genKer(DiffAlgElement,DiffAlgElement) -- see genKer -- basis of the kernel of a linear expression
- homogenize(DiffAlgElement) -- homogenize a differential form or vector field
- isHomogeneous(DiffAlgElement) -- tests if a form (or field) is homogeneous
- isInvolutive(DiffAlgDistribution) -- see isInvolutive -- tests if a distribution is involutive
- linearComb(List,String) -- see linearComb -- generic linear combination of elements
- List * DiffAlgForm -- pull-back of a differential form by a rational map
- logarithmicForm(ZZ,List,String) -- see logarithmicForm -- creates a logarithmic form
- moduliIdeal(DiffAlgElement) -- see moduliIdeal -- ideal generated by the coefficients of a differential form or vector field
- newField(ZZ,ZZ,String) -- see newField -- constructor of a vector field
- newField(String) -- constructor of a vector field
- newForm(ZZ,ZZ,ZZ,String) -- see newForm -- constructor of a differential form
- newForm(String) -- constructor of a differential form
- projectivize(DiffAlgElement) -- see projectivize -- projectivize a differential form or vector field
- radial(ZZ) -- see radial -- defines the radial vector field
- random(DiffAlgElement) -- see random(DiffAlgElement,Ring) -- replaces the variables of the coefficient ring of a differential form or a vector field with random values
- random(DiffAlgElement,Ring) -- replaces the variables of the coefficient ring of a differential form or a vector field with random values
- rank(DiffAlgDistribution) -- rank of the given distribution
- ring(DiffAlgElement) -- ring of the differential form or vector field
- singularIdeal(DiffAlgElement) -- see singularIdeal -- ideal generated by the polynomial coefficients of a differential form or vector field
- substitute(DiffAlgElement,Ring) -- gets the RingElement of a differential form or vector field in a ring

For the programmer

The object DiffAlg is a package.