From basic vector algebra we know the initial relationship between the determinant and the area (3.3-10), which might alternatively be expressed by the cross product. Note, that the cross product is defined as a scalar in two space dimensions [Dir86].
The general differential increment of a position vector is given by
Consequently, an increment of arc length along a general
space curve then is
The dot products (, ...) are known as elements
of the covariant metric tensor. An increment of arc length
on a
coordinate along which just
or just
varies is given by
(3.3-13).
We define the differential surface element as the vector of
length
directed outward normal to the surface. We recognize the
contravariant base vectors
and
as vectors normal to
coordinate surfaces
and
, respectively, so we
get (3.3-14) and (3.3-15) for the
differential surface element
and
, respectively.
This rudimentary discussion of metric and differential geometric properties is sufficient for our purposes. Further discussion can be found in virtually any text on differential geometry (e.g. [Bur79], [Sim82], [Cra86], [Abr88]).