Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

contrary is specifically stated). All other aspects of the summation convention

remain unchanged.

With the introduction of superscripts, the reciprocity relation (26.53) should be

rewritten so that both sides of (26.54) have one subscript and one superscript, i.e.

as

ei·ej=δij. (26.54)

The alternative form of the Kronecker delta is defined in a similar way to

previously, i.e. it equals unity ifi=jand is zero otherwise.

For similar reasons it is usual to denote the curvilinear coordinates themselves

byu^1 ,u^2 ,u^3 , with the index raised, so that

ei=

∂r

∂ui

and ei=∇ui. (26.55)

From the first equality we see that we may consider a superscript that appears in

the denominator of a partial derivative as a subscript.

Given the two baseseiandei, we may write a general vectoraequally well in

terms of either basis as follows:

a=a^1 e 1 +a^2 e 2 +a^3 e 3 =aiei;

a=a 1 e^1 +a 2 e^2 +a 3 e^3 =aiei.

The aiare called the contravariantcomponents of the vectoraand the ai

thecovariantcomponents, the position of the index (either as a subscript or

superscript) serving to distinguish between them. Similarly, we may call theeithe

covariant basis vectors and theeithe contravariant ones.

Show that the contravariant and covariant components of a vectoraare given byai=a·ei

andai=a·eirespectively.

For the contravariant components, we find

a·ei=ajej·ei=ajδij=ai,

where we have used the reciprocity relation (26.54). Similarly, for the covariant components,

a·ei=ajej·ei=ajδji=ai.

The reason that the notion of contravariant and covariant components of

a vector (and the resulting superscript notation) was not introduced earlier is

that for Cartesian coordinate systems the two sets of basis vectorseiandeiare

identical and, hence, so are the components of a vector with respect to either

basis. Thus, for Cartesian coordinates, we may speak simply of the components

of the vector and there is no need to differentiate between contravariance and

covariance, or to introduce superscripts to make a distinction between them.

If we consider the components of higher-order tensors in non-Cartesian co-

ordinates, there are even more possibilities. As an example, let us consider a