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D.15.12 difform_lib

Procedures for differential forms
Peter Chini, chini@rhrk.uni-kl.de

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.


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