Editing Tensor contraction (section)

== Contraction in index notation ==

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

: <math> \tilde f (\vec v) = f_\gamma v^\gamma, </math>

which is shorthand for the explicit coordinate summation<ref name="physics">In physics (and sometimes in mathematics), indices often start with zero instead of one. In four-dimensional spacetime, indices run from 0 to 3.</ref>

: <math> f_\gamma v^\gamma = f_1 v^1 + f_2 v^2 + \cdots + f_n v^n </math>

(where {{math|''v''<sup>''i''</sup>}} are the components of {{math|''v''}} in a particular basis and {{math|''f''<sub>''i''</sub>}} are the components of {{math|''f''}} in the corresponding dual basis).

Since a general mixed [[dyadic tensor]] is a linear combination of decomposable tensors of the form <math>f \otimes v</math>, the explicit formula for the dyadic case follows: let

: <math> \mathbf{T} = T_{j}^i \mathbf{e}_i \otimes \mathbf{e}^j </math>

be a mixed dyadic tensor. Then its contraction is

: <math> T_{j}^i  \mathbf{e}_i \cdot \mathbf{e}^j = T_{j}^i  \delta_i {}^j
= T_{j}^j= T_{1}^1 + \cdots + T_{n}^n </math>.

A general contraction is denoted by labeling one [[Covariance and contravariance of vectors|covariant]] index and one [[Covariance and contravariance of vectors|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

: <math> T^{ab} {}_{bc} = \sum_{b}{T^{ab}{}_{bc}} = T^{a1} {}_{1c} + T^{a2} {}_{2c} + \cdots + T^{an} {}_{nc} = U^a {}_c .</math>

By contrast, let

: <math> \mathbf{T} = \mathbf{e}^i \otimes \mathbf{e}^j </math>

be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted,{{clarification|What is "dotted" supposed to mean here? Since it is not a contraction, as is explicitly stated, then what is its definition? In particular, how is the tensor <math>g^{ij}</math> intended to be different from the tensor <math>\mathbf{T}</math>? Right now the difference only looks formal, i.e. different notation for what must otherwise be the same object.|date=May 2020}} the result is the contravariant [[metric (mathematics)|metric tensor]],

: <math> g^{ij} = \mathbf{e}^i \cdot \mathbf{e}^j </math>,

whose rank is 2.