Two-point tensor

As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates. Thus, a two-point tensor will have one capital and one lower-case index; for example, A_jM.

Continuum mechanics

A conventional tensor can be viewed as a transformation of vectors in one coordinate system to other vectors in the same coordinate system. In contrast, a two-point tensor transforms vectors from one coordinate system to another. That is, a conventional tensor,

${\displaystyle \mathbf{Q}=Q_{pq}({\mathbf{e}}_p\otimes {\mathbf{e}}_q)}$,

actively transforms a vector u to a vector v such that

${\displaystyle \mathbf{v}=\mathbf{Q}\mathbf{u}}$

where v and u are measured in the same space and their coordinates representation is with respect to the same basis (denoted by the "e").

In contrast, a two-point tensor, G will be written as

${\displaystyle \mathbf{G}=G_{pq}({\mathbf{e}}_p\otimes {\mathbf{E}}_q)}$

and will transform a vector, U, in E system to a vector, v, in the e system as

${\displaystyle \mathbf{v}=\mathbf{G}\mathbf{U}}$.