# DeRhamComplex(CoefRing, listIndVar)¶

The deRham complex of Euclidean space, that is, the class of differential forms of arbitrary 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 `n`th basis term for a differential form.

homogeneous?: % -> Boolean

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

latex: % -> String

from SetCategory

leadingBasisTerm: % -> %

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

leadingCoefficient: % -> Expression CoefRing

`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