ai1, ⋯, id for ij = 1 ⋯ nj for j = 1 ⋯ d.
A scalar is a 0-tensor. A vector is a 1-tensor. A matrix is a 2-tensor.
Alternating Multilinear Map
Penrose Graphical Notation
The covariant metric tensor, contracted with a (m, n)-tensor, yields a (m − 1, n + 1)-tensor, whereas its contravariant inverse, contracted with a (m, n)-tensor, yields a (m + 1, n − 1)-tensor.