# DifferentialExtension RΒΆ

- R: Ring

Differential extensions of a ring `R`

. Given a differentiation on `R`

, extend it to a differentiation on %.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (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: (%, %) -> %
- from NonAssociativeSemiRng
- associator: (%, %, %) -> %
- from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Integer -> %
- from NonAssociativeRing
- commutator: (%, %) -> %
- from NonAssociativeRng
- D: % -> % if R has DifferentialRing
- from DifferentialRing
- D: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from 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
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: % -> % if R has DifferentialRing
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from 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
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if R has PartialDifferentialRing Symbol
- from PartialDifferentialRing Symbol
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- 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)
- from CancellationAbelianMonoid
- zero?: % -> Boolean
- from AbelianMonoid

BiModule(%, %)

DifferentialRing if R has DifferentialRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol