In principle, expressions for the differential operators, such as gradient
(
or 
), divergence (
or 
), curl
(
or 
or 
) and Laplacian
(
), can be obtained by inserting the expressions
(3.3-44) into the operators in cartesian coordinates. The major
drawback of this attempt is that global conservation properties implied by
the resultant equations cannot be seen immediately. However, these
properties play an important role when a discretization scheme for the
equations is sought.
Alternatively, we deduce the expressions for the differential operators
from the divergence theorem (generalized Gauss' law) (3.3-46)
which is valid for a tensor 
of any rank.
We apply its two-dimensional form
to an area element 
bounded by four segments lying on coordinate lines
, 
, 
, 
. Keeping in mind the definition of 
(3.3-14) and 
(3.3-15), we
obtain
Proceeding to the limit as the element's area shrinks to zero (
), we then have an expression for the divergence of a vector
(3.3-50).
As the divergence theorem (3.3-47) is valid for a tensor
of any rank, we can apply (3.3-48) to a scalar
valued function 
to get an expression for the gradient of
(3.3-51).
The alternative form of the expressions for the differential operators
derived from the chain rule look quite similar. The two formulations are
even mathematically identical if we recall the metric identity 
.
Although the equations (3.3-54) and (3.3-52) are mathematically equivalent formulations for the divergence operator, the numerical representations of these two forms are not equivalent.
We called the form given by (3.3-50) conservative form,
and call that of (3.3-52) the non-conservative form of
the differential operators. Recalling that the quantity 
represents a surface element (Figure 3.3-6, so that
is a flux through this element, it is clear
that the difference between the two forms is that the surface element used
in the numerical representation of the flux in conservative form
(3.3-50) is the surface element of the individual sides of the
area element , but in non-conservative form a common surface element at
the center of the area element is used. The conservative form thus gives the
telescopic collapse of the flux terms when the difference approximations are
summed-up over the field, so that this summation then involves only the
boundary fluxes. The global conservation properties of the conservative form
are superior due to the better numerical representation of the net flux through
an area element.
Equations (3.3-54) and (3.3-55) resemble the conservative forms of the operators, which are used in PROMIS.
