Tensor Contraction - Contraction in Index Notation

Contraction in Index Notation

In abstract index notation, the basic contraction of a vector and a dual vector is denoted by

which is shorthand for the explicit coordinate summation

(where vi are the components of v in a particular basis and f_i are the components of f in the corresponding dual basis).

Since a general mixed dyadic tensor is a linear combination of decomposable tensors of the form, the explicit formula for the dyadic case follows: let

be a mixed dyadic tensor. Then its contraction is

.

A general contraction is denoted by labeling one covariant index and one contravariant index with the same letter, summation over that index being implied by the summation convention. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type (2,2) on the second and third indices to create a new tensor U of type (1,1) is written as

By contrast, let

be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor,

,

whose rank is 2.