# DeRhamComplex(CoefRing, listIndVar)¶

The deRham complex of Euclidean space, that is, the class of differential forms of arbitary degree over a coefficient ring. See Flanders, Harley, Differential Forms, With Applications to the Physical Sciences, New York, Academic Press, 1963.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (Expression CoefRing, %) -> %

from LeftModule Expression CoefRing

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coefficient: (%, %) -> Expression CoefRing

coefficient(df, u), where df is a differential form, returns the coefficient of df containing the basis term u if such a term exists, and 0 otherwise.

coerce: % -> OutputForm
coerce: Expression CoefRing -> %

from LeftAlgebra Expression CoefRing

coerce: Integer -> %
commutator: (%, %) -> %
degree: % -> NonNegativeInteger

degree(df) returns the homogeneous degree of differential form df.

exteriorDifferential: % -> %

exteriorDifferential(df) returns the exterior derivative (gradient, curl, divergence, …) of the differential form df.

generator: NonNegativeInteger -> %

generator(n) returns the nth basis term for a differential form.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

homogeneous?: % -> Boolean

homogeneous?(df) tests if all of the terms of differential form df have the same degree.

latex: % -> String

from SetCategory

leadingBasisTerm(df) returns the leading basis term of differential form df.

leadingCoefficient(df) returns the leading coefficient of differential form df.

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

map: (Expression CoefRing -> Expression CoefRing, %) -> %

map(f, df) replaces each coefficient x of differential form df by f(x).

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

reductum(df), where df is a differential form, returns df minus the leading term of df if df has two or more terms, and 0 otherwise.

retract: % -> Expression CoefRing

from RetractableTo Expression CoefRing

retractable?: % -> Boolean

retractable?(df) tests if differential form df is a 0-form, i.e. if degree(df) = 0.

retractIfCan: % -> Union(Expression CoefRing, failed)

from RetractableTo Expression CoefRing

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)
totalDifferential: Expression CoefRing -> %

totalDifferential(x) returns the total differential (gradient) form for element x.

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CoercibleFrom Expression CoefRing

LeftAlgebra Expression CoefRing

LeftModule Expression CoefRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

RetractableTo Expression CoefRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown