The operator 
represents the mapping of a specific point 
of the computational domain to a point 
in the physical domain
(3.3-2). It is worth noting that in general is a
nonlinear mapping of Banach spaces and differentiability is expressed as
Fréchet-differentiability [Dir86]. The transformation is
accomplished by the set of functions 
and 
(3.3-3).
In the further text coordinates appearing as subscripts indicate partial
differentiation, for instance 
.
In two dimensions the index 1 is associated with 
and 
. The index 2 is
associated with 
and 
. The Jacobian 
of the transformation
is given by (3.3-4), 
's
Fréchet-derivative 
.
gives the differential coordinate transformation
(3.3-5), and consequently the shape of the distorted cell in the
physical space that corresponds to an infinitesimal square cell in
computational space.
The Jacobian's determinant 
measures the area of the distorted cell in
the physical space.
For non-vanishing determinants the inverse of exists and a
one-to-one mapping can be established. These mappings are called regular. We prefer a sloppy but more practical and comprehensible
interpretation of regularity [Son89]:
A grid is called regular if all grid lines lie in the closures of the physical domain and if there is no intersection of grid lines of the same family and if two lines of different families intersect only once.
