Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

where the elementsLijare given by

L=





cosθ sinθ 0

−sinθ cosθ 0

001



.

Thus (26.68) and (26.70) agree with our earlier definition in the special case of a

rigid rotation of Cartesian axes.

Following on from (26.68) and (26.70), we proceed in a similar way to de-

fine general tensors of higher rank. For example, the contravariant, mixed and

covariant components, respectively, of a second-order tensor must transform as

follows:

contravariant components, T′

ij

=

∂u′i

∂uk

∂u′j

∂ul

Tkl;

mixed components, T′

i

j=

∂u′i

∂uk

∂ul

∂u′j

Tkl;

covariant components, T′ij=

∂uk

∂u′i

∂ul

∂u′j

Tkl.

It is important to remember that these quantities form the components of the

sametensorTbut refer to different tensor bases made up from the basis vectors

of the different coordinate systems. For example, in terms of the contravariant

components we may write

T=Tijei⊗ej=T′

ij

e′i⊗e′j.

We can clearly go on to define tensors of higher order, with arbitrary numbers

of covariant (subscript) and contravariant (superscript) indices, by demanding

that their components transform as follows:

T′ij···klm···n=

∂u′i

∂ua

∂u′j

∂ub

···

∂u′k

∂uc

∂ud

∂u′l

∂ue

∂u′m

···

∂uf

∂u′n

Tab···cde···f. (26.71)

Using the revised summation convention described in section 26.14, the algebra

of general tensors is completely analogous to that of the Cartesian tensors

discussed earlier. For example, as with Cartesian coordinates, the Kronecker

delta is a tensor provided it is written as the mixed tensorδjisince

δ′

i

j=

∂u′i

∂uk

∂ul

∂u′j

δkl=

∂u′i

∂uk

∂uk

∂u′j

=

∂u′i

∂u′j

=δji,

where we have used the chain rule to justify the third equality. This also shows

thatδij is isotropic. As discussed at the end of section 26.15, theδjican be

considered as the mixed components of the metric tensorg.