# DifferentialExtension RΒΆ

catdef.spad line 268 [edit on github]

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 LeftModule %

- *: (Integer, %) -> %
from AbelianGroup

- *: (NonNegativeInteger, %) -> %
from AbelianMonoid

- *: (PositiveInteger, %) -> %
from AbelianSemiGroup

- +: (%, %) -> %
from AbelianSemiGroup

- -: % -> %
from AbelianGroup

- -: (%, %) -> %
from AbelianGroup

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- 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
- 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

BiModule(%, %)

DifferentialRing if R has DifferentialRing

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol