Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.5 Second- and higher-order Cartesian tensors

Ifviare the components of a first-order tensor, show that∇·v=∂vi/∂xiis a zero-order

tensor.

In the rotated coordinate system∇·vis given by

(

∂vi

∂xi

)′

=

∂vi′

∂x′i

=

∂xj

∂x′i

∂

∂xj

(Likvk)=LijLik

∂vk

∂xj

,

since the elementsLijare not functions of position. Using the orthogonality relation (26.6)

we then find

∂v′i

∂x′i

=LijLik

∂vk

∂xj

=δjk

∂vk

∂xj

=

∂vj

∂xj

.

Hence∂vi/∂xiis invariant under rotation of the axes and is thus a zero-order tensor; this

was to be expected since it can be written in a coordinate-free way as∇·v.

26.5 Second- and higher-order Cartesian tensors

Following on from scalars with no subscripts and vectors with one subscript,

we turn to sets of quantities that require two subscripts to identify a particular

element of the set. Let these quantities by denoted byTij.

Taking (26.9) as a guide we define asecond-order Cartesian tensoras follows:

theTijform the components of such a tensor if, under the same conditions as

for (26.9),

Tij′=LikLjlTkl (26.16)

and

Tij=LkiLljTkl′. (26.17)

At the same time we may define a Cartesian tensor of general order as follows.

The set of expressionsTij···kform the components of a Cartesian tensor if, for all

rotations of the axes of coordinates given by (26.4) and (26.5), subject to (26.6),

the expressions using the new coordinates,Tij′···kare given by

Tij′···k=LipLjq···LkrTpq···r (26.18)

and
Tij···k=LpiLqj···LrkTpq′···r. (26.19)

It is apparent that in three dimensions, anNth-order Cartesian tensor has 3N

components.

Since a second-order tensor has two subscripts, it is natural to display its

components in matrix form. The notation [Tij] is used, as well asT, to denote

the matrix havingTijas the element in theith row andjth column.§

We may think of a second-order tensorTas a geometrical entity in a similar

way to that in which we viewed linear operators (which transform one vector into

§We can also denote the column matrix containing the elementsviof a vector by [vi].