# DifferentialExtension RΒΆ

Differential extensions of a ring `R`. Given a differentiation on `R`, extend it to a differentiation on %.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (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
coerce: % -> OutputForm
coerce: Integer -> %
commutator: (%, %) -> %
D: % -> % if R has DifferentialRing

from DifferentialRing

D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

D: (%, R -> R) -> %

`D(x, deriv)` differentiates `x` extending the derivation deriv on `R`.

D: (%, R -> R, NonNegativeInteger) -> %

`D(x, deriv, n)` differentiate `x` `n` times using a derivation which extends `deriv` on `R`.

D: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
differentiate: % -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if R has DifferentialRing

from DifferentialRing

differentiate: (%, R -> R) -> %

`differentiate(x, deriv)` differentiates `x` extending the derivation deriv on `R`.

differentiate: (%, R -> R, NonNegativeInteger) -> %

`differentiate(x, deriv, n)` differentiate `x` `n` times using a derivation which extends `deriv` on `R`.

differentiate: (%, Symbol) -> % if R has PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

subtractIfCan: (%, %) -> Union(%, failed)
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

DifferentialRing if R has DifferentialRing

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown