In order to make use of a general boundary conforming curvilinear coordinate
system in the solution of partial differential equations, the equations must
first be transformed to the curvilinear coordinates. Such a transformation
is accomplished by means of the relations developed in
Section 3.3.4. It produces a problem where the independent
variables are time 
and the curvilinear coordinates 
and 
. The
resulting equations are of the same type as the original ones but are more
complicated as they contain more terms. The domain where these
equations have to be solved, on the other hand, is greatly simplified. It is
a fixed rectangular region regardless of the shape and movement of the
physical domain. This facilitates the imposition of boundary conditions and
is the primary feature which makes curvilinear grids a valuable tool for
solving PDEs on a nonplanar domain.
We use the conservative formulations for the transformation of the PDEs. Introducing (3.3-54), (3.3-55), (3.3-57), (3.3-59) and (3.3-60) into (3.1-1), (3.1-2) and (3.1-3) yields after some arithmetics the continuity equation (3.3-61), the fluxes (3.3-62) and (3.3-63), and the boundary conditions (3.3-64) and (3.3-65).
Boundary conditions for the north 
and the south 
boundary are
resembled in (3.3-64), and those for the west and the east
boundary are given in (3.3-65).
The expressions (3.3-61) - (3.3-65) are the equations
which have to be solved in the computational domain 
.
