# PartialDifferentialRing SΒΆ

- S: SetCategory

A partial differential ring with differentiations indexed by a parameter type `S`

.

- 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: (%, List S) -> %
`D(x, [s1, ...sn])`

computes successive partial derivatives, i.e.`D(...D(x, s1)..., sn)`

.

- D: (%, List S, List NonNegativeInteger) -> %
`D(x, [s1, ..., sn], [n1, ..., nn])`

computes multiple partial derivatives, i.e.`D(...D(x, s1, n1)..., sn, nn)`

.

- D: (%, S) -> %
`D(x, v)`

computes the partial derivative of`x`

with respect to`v`

.

- D: (%, S, NonNegativeInteger) -> %
`D(x, s, n)`

computes multiple partial derivatives, i.e.`n`

-th derivative of`x`

with respect to`s`

.

- differentiate: (%, List S) -> %
`differentiate(x, [s1, ...sn])`

computes successive partial derivatives, i.e.`differentiate(...differentiate(x, s1)..., sn)`

.

- differentiate: (%, List S, List NonNegativeInteger) -> %
`differentiate(x, [s1, ..., sn], [n1, ..., nn])`

computes multiple partial derivatives, i.e.

- differentiate: (%, S) -> %
`differentiate(x, v)`

computes the partial derivative of`x`

with respect to`v`

.

- differentiate: (%, S, NonNegativeInteger) -> %
`differentiate(x, s, n)`

computes multiple partial derivatives, i.e.`n`

-th derivative of`x`

with respect to`s`

.- 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(%, %)