CartesianTensor(minix, dim, R)ΒΆ

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CartesianTensor(minix, dim, R) provides Cartesian tensors with components belonging to a commutative ring R. These tensors can have any number of indices. Each index takes values from minix to minix + dim - 1.

0: %

from GradedModule(Integer, NonNegativeInteger)

1: %

from GradedAlgebra(R, NonNegativeInteger)

*: (%, %) -> %

s*t is the inner product of the tensors s and t which contracts the last index of s with the first index of t, i.e. t*s = contract(t, rank t, s, 1) t*s = sum(k=1..N, t[i1, .., iN, k]*s[k, j1, .., jM]) This is compatible with the use of M*v to denote the matrix-vector inner product.

*: (%, Integer) -> %

from GradedModule(Integer, NonNegativeInteger)

*: (%, R) -> %

from GradedModule(R, NonNegativeInteger)

*: (Integer, %) -> %

from GradedModule(Integer, NonNegativeInteger)

*: (R, %) -> %

from GradedModule(R, NonNegativeInteger)

+: (%, %) -> %

from GradedModule(Integer, NonNegativeInteger)

-: % -> %

from GradedModule(Integer, NonNegativeInteger)

-: (%, %) -> %

from GradedModule(Integer, NonNegativeInteger)

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: DirectProduct(dim, R) -> %

coerce(v) views a vector as a rank 1 tensor.

coerce: List % -> %

coerce([t_1, ..., t_dim]) allows tensors to be constructed using lists.

coerce: List R -> %

coerce([r_1, ..., r_dim]) allows tensors to be constructed using lists.

coerce: R -> %

from CoercibleFrom R

coerce: SquareMatrix(dim, R) -> %

coerce(m) views a matrix as a rank 2 tensor.

contract: (%, Integer, %, Integer) -> %

contract(t, i, s, j) is the inner product of tenors s and t which sums along the k1-th index of t and the k2-th index of s. For example, if r = contract(s, 2, t, 1) for rank 3 tensors rank 3 tensors s and t, then r is the rank 4 (= 3 + 3 - 2) tensor given by r(i, j, k, l) = sum(h=1..dim, s(i, h, j)*t(h, k, l)).

contract: (%, Integer, Integer) -> %

contract(t, i, j) is the contraction of tensor t which sums along the i-th and j-th indices. For example, if r = contract(t, 1, 3) for a rank 4 tensor t, then r is the rank 2 (= 4 - 2) tensor given by r(i, j) = sum(h=1..dim, t(h, i, h, j)).

degree: % -> NonNegativeInteger

from GradedModule(Integer, NonNegativeInteger)

elt: % -> R

elt(t) gives the component of a rank 0 tensor.

elt: (%, Integer) -> R

elt(t, i) gives a component of a rank 1 tensor.

elt: (%, Integer, Integer) -> R

elt(t, i, j) gives a component of a rank 2 tensor.

elt: (%, Integer, Integer, Integer) -> R

elt(t, i, j, k) gives a component of a rank 3 tensor.

elt: (%, Integer, Integer, Integer, Integer) -> R

elt(t, i, j, k, l) gives a component of a rank 4 tensor.

elt: (%, List Integer) -> R

elt(t, [i1, ..., iN]) gives a component of a rank N tensor.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

kroneckerDelta: () -> %

kroneckerDelta() is the rank 2 tensor defined by kroneckerDelta()(i, j) = 1 if i = j = 0 if i \~= j

latex: % -> String

from SetCategory

leviCivitaSymbol: () -> %

leviCivitaSymbol() is the rank dim tensor defined by leviCivitaSymbol()(i1, ...idim) = +1/0/-1 if i1, ..., idim is an even/is nota /is an odd permutation of minix, ..., minix+dim-1.

product: (%, %) -> %

product(s, t) is the outer product of the tensors s and t. For example, if r = product(s, t) for rank 2 tensors s and t, then r is a rank 4 tensor given by r(i, j, k, l) = s(i, j)*t(k, l).

rank: % -> NonNegativeInteger

rank(t) returns the tensorial rank of t (that is, the number of indices). This is the same as the graded module degree.

ravel: % -> List R

ravel(t) produces a list of components from a tensor such that unravel(ravel(t)) = t.

reindex: (%, List Integer) -> %

reindex(t, [i1, ..., idim]) permutes the indices of t. For example, if r = reindex(t, [4, 1, 2, 3]) for a rank 4 tensor t, then r is the rank for tensor given by r(i, j, k, l) = t(l, i, j, k).

retract: % -> R

from RetractableTo R

retractIfCan: % -> Union(R, failed)

from RetractableTo R

sample: () -> %

sample() returns an object of type %.

transpose: % -> %

transpose(t) exchanges the first and last indices of t. For example, if r = transpose(t) for a rank 4 tensor t, then r is the rank 4 tensor given by r(i, j, k, l) = t(l, j, k, i).

transpose: (%, Integer, Integer) -> %

transpose(t, i, j) exchanges the i-th and j-th indices of t. For example, if r = transpose(t, 2, 3) for a rank 4 tensor t, then r is the rank 4 tensor given by r(i, j, k, l) = t(i, k, j, l).

unravel: List R -> %

unravel(t) produces a tensor from a list of components such that unravel(ravel(t)) = t.

BasicType

CoercibleFrom R

CoercibleTo OutputForm

GradedAlgebra(R, NonNegativeInteger)

GradedModule(Integer, NonNegativeInteger)

GradedModule(R, NonNegativeInteger)

RetractableTo R

SetCategory