Two-point tensor

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"Double vector" redirects here. For dual vectors, see dual space. For bivectors, see bivector.

Two-point tensors, or double vectors, are tensor-like quantities which transform as vectors with respect to each of their indices and are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates.[1] Examples include the deformation gradient and the first Piola–Kirchhoff stress 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, AjM.

Continuum mechanics[edit]

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,

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

actively transforms a vector u to a vector v such that


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

\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


The transformation law for two-point tensor[edit]

Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as


For tensors suppose we then have

T_{pq}(e_p \otimes e_q).

A tensor in the system e_i. In another system, let the same tensor be given by

T'_{pq}(e'_p \otimes e'_q).

We can say

T'_{ij}=Q_{ip} Q_{jr} T_{pr}.



is the routine tensor transformation. But a two-point tensor between these systems is just

F_{pq}(e'_p \otimes e_q)

which transforms as


The most mundane example of a two-point tensor[edit]

The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that


Now, writing out in full,

u=u_q e_q

and also

v=v'_p e_p.

This then requires Q to be of the form

Q_{pq}(e'_p \otimes e_q).

By definition of tensor product,

(e'_p\otimes e_q)e_q=(e_q.e_q) e'_p = e'_p\qquad(1)

So we can write

u_p e_p = (Q_{pq}(e'_p \otimes e_q))(v_q e_q)


u_p e_p = Q_{pq} v_q(e'_p \otimes e_q) e_q

Incorporating (1), we have

u_p e_p = Q_{pq} v_q e_p.

See also[edit]

External links[edit]


  1. ^ Humphrey, Jay D. Cardiovascular solid mechanics: cells, tissues, and organs. Springer Verlag, 2002.