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]).
