NonCartesian coordinates

26.14 NON-CARTESIAN COORDINATES
The other integral theorems discussed in chapter 11 can be extended in a
similar way. For example, written in tensor notation Stokes’ theorem states that,
for a vector field ai ,
0
∂ak
n̂i dS =
ijk
ak dxk .
∂xj
S
C
For a general tensor field this has the straightforward extension
0
∂Tlm···k···n
n̂i dS =
ijk
Tlm···k···n dxk .
∂xj
S
C
26.14 Non-Cartesian coordinates
So far we have restricted our attention to the study of tensors when they are
described in terms of Cartesian coordinates and the axes of coordinates are rigidly
rotated, sometimes together with an inversion of axes through the origin. In the
remainder of this chapter we shall extend the concepts discussed in the previous
sections by considering arbitrary coordinate transformations from one general
coordinate system to another. Although this generalisation brings with it several
complications, we shall find that many of the properties of Cartesian tensors
are still valid for more general tensors. Before considering general coordinate
transformations, however, we begin by reminding ourselves of some properties of
general curvilinear coordinates, as discussed in chapter 10.
The position of an arbitrary point P in space may be expressed in terms of the
three curvilinear coordinates u1 , u2 , u3 . We saw in chapter 10 that if r(u1 , u2 , u3 ) is
the position vector of the point P then at P there exist two sets of basis vectors
ei =
∂r
∂ui
and
i = ∇ui ,
(26.52)
where i = 1, 2, 3. In general, the vectors in each set neither are of unit length nor
form an orthogonal basis. However, the sets ei and i are reciprocal systems of
vectors and so
ei · j = δij .
(26.53)
In the context of general tensor analysis, it is more usual to denote the second
set of vectors i in (26.52) by ei , the index being placed as a superscript to
distinguish it from the (different) vector ei , which is a member of the first set in
(26.52). Although this positioning of the index may seem odd (not least because
of the possibility of confusion with powers) it forms part of a slight modification
to the summation convention that we will adopt for the remainder of this chapter.
This is as follows: any lower-case alphabetic index that appears exactly twice in
any term of an expression, once as a subscript and once as a superscript, is to be
summed over all the values that an index in that position can take (unless the
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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 · e j = δij .
(26.54)
The alternative form of the Kronecker delta is defined in a similar way to
previously, i.e. it equals unity if i = j and is zero otherwise.
For similar reasons it is usual to denote the curvilinear coordinates themselves
by u1 , u2 , u3 , with the index raised, so that
∂r
and
ei = ∇ui .
(26.55)
∂ui
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 bases ei and ei , we may write a general vector a equally well in
terms of either basis as follows:
ei =
a = a1 e1 + a2 e2 + a3 e3 = ai ei ;
a = a1 e1 + a2 e2 + a3 e3 = ai ei .
The ai are called the contravariant components of the vector a and the ai
the covariant components, the position of the index (either as a subscript or
superscript) serving to distinguish between them. Similarly, we may call the ei the
covariant basis vectors and the ei the contravariant ones.
Show that the contravariant and covariant components of a vector a are given by ai = a·ei
and ai = a · ei respectively.
For the contravariant components, we find
a · ei = a j ej · ei = a j δji = ai ,
where we have used the reciprocity relation (26.54). Similarly, for the covariant components,
a · ei = aj e j · ei = aj δij = 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 vectors ei and ei are
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 coordinates, there are even more possibilities. As an example, let us consider a
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