# PartialDifferentialRing SΒΆ

catdef.spad line 1167 [edit on github]

S: SetCategory

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

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

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

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

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

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

computes the partial derivative of`x`

with respect to`v`

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

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

-th derivative of`x`

with respect to`s`

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

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

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- 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`

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- 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(%, %)