Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

another, without reference to any coordinate system) and consider the matrix

containing its components as a representation of the tensor with respect to a

particular coordinate system. Moreover, the matrixT=[Tij], containing the

components of a second-order tensor, behaves in the same way under orthogonal

transformationsT′=LTLTas a linear operator.

However, not all linear operators are second-order tensors. More specifically,

the two subscripts in a second-order tensor must refer to the same coordinate

system. In particular, this means that any linear operator that transforms a vector

into a vector in a different vector space cannot be a second-order tensor. Thus,

although the elementsLijof the transformation matrix are written with two

subscripts, they cannot be the components of a tensor since the two subscripts

each refer to a different coordinate system.

As examples of sets of quantities that are readily shown to be second-order

tensors we consider the following.

(i)The outer product of two vectors.Letuiandvi,i=1, 2 ,3, be the components

of two vectorsuandv, and consider the set of quantitiesTijdefined by

Tij=uivj. (26.20)

The setTijare called the components of the theouter productofuandv. Under

rotations the componentsTijbecome

Tij′=ui′v′j=LikukLjlvl=LikLjlukvl=LikLjlTkl, (26.21)

which shows that they do transform as the components of a second-order tensor.

Use has been made in (26.21) of the fact thatuiandviare the components of

first-order tensors.

The outer product of two vectors is often denoted, without reference to any

coordinate system, as

T=u⊗v. (26.22)

(This is not to be confused with the vector product of two vectors, which is itself

a vector and is discussed in chapter 7.) The expression (26.22) gives the basis to

which the componentsTijof the second-order tensor refer: sinceu=uieiand

v=viei, we may write the tensorTas

T=uiei⊗vjej=uivjei⊗ej=Tijei⊗ej. (26.23)

Moreover, as for the case of first-order tensors (see equation (26.10)) we note

that the quantitiesTij′are the components of thesametensorT, but referred to

a different coordinate system, i.e.

T=Tijei⊗ej=Tij′e′i⊗e′j.

These concepts can be extended to higher-order tensors.