Mathematica's uniform representation of vectors and matrices as lists automatically extends to tensors of any rank, allowing
Mathematica's powerful list manipulation functions immediately to be applied to tensors, both numerical and symbolic.
Table — construct a tensor of any rank from an expression
Array — construct a tensor from a function:
Array[f, dims]
SparseArray — specify a tensor in a sparse
position
value form
Dimensions — the dimensions of a tensor
ArrayDepth — the rank of a tensor
ArrayQ — test whether an object is a tensor of a given rank
MatrixForm — display a tensor of any rank
KroneckerDelta — identity tensor
Signature — elements of totally antisymmetric tensor
Band — specify banded structure in a sparse array
Transpose — transpose to rearrange indices in any way
Inner — generalized inner product
Outer — generalized outer product
Flatten — flatten out any sequence of levels
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