### D.15.4 difform_lib

Library:
  difform.lib
Purpose:
  Procedures for differential forms
Author:
  Peter Chini, chini@rhrk.uni-kl.de

Overview:
  A library for computing with elements of the differential algebra over a (quotient) ring. To compute in this algebra, a non-commutative ring with additional variables dx_1,...,dx_n and 'exterior' relations between this variables is used. In the case of a quotient ring, the defining ideal and its image under the universal derivation are added as relations. The differential forms themselves are defined via an additional type 'difform'. Objects of this type carry as an attribute a polynomial in the differential algebra and make it available over the basering.
  Additionally, the universal derivation is available as a procedure and the differentials between the graded parts of the differential algebra can be applied to differential forms. The library also supports derivations: maps from the first graded part of the differential algebra to the basering. These are defined via the type 'derivation' and there are procedures for basic arithmetic operations, evaluation and Lie-derivative.

Procedures:

> D.15.4.1 diffAlgebra |  | provides the differential algebra structure and the differential forms dx_1,...,dx_n
> D.15.4.2 diffAlgebraStructure |  | generates the structure of the differential algebra from the basering
> D.15.4.3 diffAlgebraGens |  | defines the differential forms dx_1,...,dx_n
> D.15.4.4 diffAlgebraUnivDerIdeal |  | computes the image of an ideal under the universal derivation
> D.15.4.5 diffAlgebraChangeOrd |  | returns a ring with the structure of the differential algebra but changed monomial ordering
> D.15.4.6 diffAlgebraListGen |  | returns a list of the generators of the differential algebra or of a graded part of it
> D.15.4.7 difformFromPoly |  | constructs differential forms of degree 0 from polynomials
> D.15.4.8 difformCoef |  | computes the representation as an linear combination of the generators
> D.15.4.9 difformGenToString |  | casts a generator of the differential algebra to a string
> D.15.4.10 difformHomogDecomp |  | list of differential forms: homogeneous decomposition
> D.15.4.11 difformToString |  | casts a differential form to a string
> D.15.4.12 difformPrint |  | prints differential forms
> D.15.4.13 difformIsGen |  | decides, whether a given differential form is a generator of the differential algebra
> D.15.4.14 difformAdd |  | adds two differential forms
> D.15.4.15 difformSub |  | subtracts one differential form from the other
> D.15.4.16 difformNeg |  | returns the negative of a differential form
> D.15.4.17 difformMul |  | multiplies two differential forms
> D.15.4.18 difformDiv |  | computes the quotient of two differential forms
> D.15.4.19 difformEqu |  | compares two differential forms
> D.15.4.20 difformNeq |  | returns the negation of comparing two differential forms
> D.15.4.21 difformIsBigger |  | tests if a given differential form is greater than another one
> D.15.4.22 difformIsSmaller |  | tests if a given differential form is smaller than another one
> D.15.4.23 difformDeg |  | returns the degree of a given differential form
> D.15.4.24 difformIsHomog |  | checks if the given differential form is homogeneous
> D.15.4.25 difformIsHomogDeg |  | checks if the given differential form is homogeneous of given degree
> D.15.4.26 difformListCont |  | checks if a given differential form is in a given list
> D.15.4.27 difformListSort |  | sorts lists of differential forms and special lists of lists
> D.15.4.28 difformUnivDer |  | computes the image of an polynomial under the universal derivation
> D.15.4.29 difformDiff |  | computes the image of an differential form under the differential
> D.15.4.30 derivationFromList |  | constructs a derivation from a given list
> D.15.4.31 derivationCheckList |  | checks the form of a given structure list for a derivation
> D.15.4.32 derivationFromPoly |  | creates a derivation from a polynomial
> D.15.4.33 derivationConstructor |  | constructs a derivation from arbitrary input
> D.15.4.34 derivationToString |  | casts a derivation to a string
> D.15.4.35 derivationPrint |  | prints a derivation
> D.15.4.36 derivationAdd |  | computes the sum of two derivations
> D.15.4.37 derivationSub |  | subtracts two derivations
> D.15.4.38 derivationNeg |  | negates a given derivation
> D.15.4.39 derivationMul |  | multiplies two derivations componentwise
> D.15.4.40 derivationEqu |  | compares two derivations
> D.15.4.41 derivationNeq |  | returns the negation of comparing two derivations
> D.15.4.42 derivationEval |  | evaluates a derivation at a given differential form of degree 1
> D.15.4.43 derivationContractionGen |  | computes the contraction and applies it to a generator
> D.15.4.44 derivationContraction |  | computes the contraction and applies it to a differential form
> D.15.4.45 derivationLie |  | returns the Lie-derivative applied to a differential form

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