Tensor contraction

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

Since the role of x is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.

In tensor index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.