# PartialDifferentialRing SΒΆ

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

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: (%, 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)
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

CancellationAbelianMonoid

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown