DeRhamComplex(CoefRing, listIndVar)

derham.spad line 278 [edit on github]

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.

* : (%, %) -> %
from Magma
* : (Expression(CoefRing), %) -> %
from LeftModule(Expression(CoefRing))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
associator : (%, %, %) -> %
from NonAssociativeRng
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
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 : Expression(CoefRing) -> %
from LeftAlgebra(Expression(CoefRing))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
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.

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 : (Mapping(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))
retractIfCan : % -> Union(Expression(CoefRing), "failed")
from RetractableTo(Expression(CoefRing))
retractable? : % -> Boolean

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

rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from AbelianMonoid
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
totalDifferential : Expression(CoefRing) -> %

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

zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

RightModule(%)

NonAssociativeSemiRng

Monoid

Ring

LeftAlgebra(Expression(CoefRing))

SemiGroup

CancellationAbelianMonoid

LeftModule(%)

MagmaWithUnit

RetractableTo(Expression(CoefRing))

BasicType

unitsKnown

NonAssociativeRing

Rng

CoercibleTo(OutputForm)

SemiRing

AbelianGroup

AbelianSemiGroup

SetCategory

CoercibleFrom(Expression(CoefRing))

AbelianMonoid

Magma

LeftModule(Expression(CoefRing))

BiModule(%, %)

NonAssociativeRng

NonAssociativeSemiRing

SemiRng