Singular

D.15.12 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.15.12.1 diffAlgebra		provides the differential algebra structure and the differential forms dx_1,...,dx_n
D.15.12.2 diffAlgebraStructure		generates the structure of the differential algebra from the basering
D.15.12.3 diffAlgebraCheck		checks if the basering already has a differential algebra
D.15.12.4 diffAlgebraSwitch		changes the basering to the differential algebra for computations
D.15.12.5 diffAlgebraGens		defines the differential forms dx_1,...,dx_n
D.15.12.6 diffAlgebraUnivDerIdeal		computes the image of an ideal under the universal derivation
D.15.12.7 diffAlgebraChangeOrd		returns a ring with the structure of the differential algebra but changed monomial ordering
D.15.12.8 diffAlgebraListGen		returns a list of the generators of the differential algebra or of a graded part of it
D.15.12.9 difformFromPoly		constructs differential forms of degree 0 from polynomials
D.15.12.10 difformCoef		computes the representation as an linear combination of the generators
D.15.12.11 difformGenToString		casts a generator of the differential algebra to a string
D.15.12.12 difformHomogDecomp		list of differential forms: homogeneous decomposition
D.15.12.13 difformToString		casts a differential form to a string
D.15.12.14 difformPrint		prints differential forms
D.15.12.15 difformIsGen		decides, whether a given differential form is a generator of the differential algebra
D.15.12.16 difformAdd		adds two differential forms
D.15.12.17 difformSub		subtracts one differential form from the other
D.15.12.18 difformNeg		returns the negative of a differential form
D.15.12.19 difformMul		multiplies two differential forms
D.15.12.20 difformDiv		computes the quotient of two differential forms
D.15.12.21 difformEqu		compares two differential forms
D.15.12.22 difformNeq		returns the negation of comparing two differential forms
D.15.12.23 difformIsBigger		tests if a given differential form is greater than another one
D.15.12.24 difformIsSmaller		tests if a given differential form is smaller than another one
D.15.12.25 difformDeg		returns the degree of a given differential form
D.15.12.26 difformIsHomog		checks if the given differential form is homogeneous
D.15.12.27 difformIsHomogDeg		checks if the given differential form is homogeneous of given degree
D.15.12.28 difformListCont		checks if a given differential form is in a given list
D.15.12.29 difformListSort		sorts lists of differential forms and special lists of lists
D.15.12.30 difformUnivDer		computes the image of an polynomial under the universal derivation
D.15.12.31 difformDiff		computes the image of an differential form under the differential
D.15.12.32 derivationFromList		constructs a derivation from a given list
D.15.12.33 derivationCheckList		checks the form of a given structure list for a derivation
D.15.12.34 derivationFromPoly		creates a derivation from a polynomial
D.15.12.35 derivationConstructor		constructs a derivation from arbitrary input
D.15.12.36 derivationToString		casts a derivation to a string
D.15.12.37 derivationPrint		prints a derivation
D.15.12.38 derivationAdd		computes the sum of two derivations
D.15.12.39 derivationSub		subtracts two derivations
D.15.12.40 derivationNeg		negates a given derivation
D.15.12.41 derivationMul		multiplies two derivations componentwise
D.15.12.42 derivationEqu		compares two derivations
D.15.12.43 derivationNeq		returns the negation of comparing two derivations
D.15.12.44 derivationEval		evaluates a derivation at a given differential form of degree 1
D.15.12.45 derivationContractionGen		computes the contraction and applies it to a generator
D.15.12.46 derivationContraction		computes the contraction and applies it to a differential form
D.15.12.47 derivationLie		returns the Lie-derivative applied to a differential form