GENERAL CURVILINEAR SYSTEMS
55
To evaluate the numerator, integration over all six surfaces of
V
must be performed.
First, consider the two shaded sides
of
Figure
3.9,
with normals aligned either parallel
or antiparallel to
q3.
The integral over the “bottom” surface is
The minus sign arises because on this surface
diF
and
$3
are antiparallel.
Also
notice
that
A3,
hl
,
and
h2
are
all
functions of the curvilinear coordinates and are evaluated
at
(41, q2, q3),
the initial values
of
the coordinates on this surface. The integral over
the “top” surface is
In this case there is no minus sign because
this
surface normal is oriented parallel
to
$3.
The initial value of the
q3
coordinate for
this
surface has changed by an amount
dq3
as compared to the bottom surface and thus
A3,
hl,
and
h2
must all be evaluated
at the point
(ql, q2, q3
+
dq3). In
the differential limit
so
the sum of Equations
3.44
and
3.45
is
(3.47)
Combining this result with similar integrations over the remaining four surfaces gives
3.4.10
The
Curl
The curl operation for
a
curvilinear coordinate system can also be derived from its
integral definition:
v
X
A.
lim
da
=
lim
dT;.
A,
s-0
c-0
(3.50)
where
C
is
a closed path surrounding the surface
S,
and the direction
of
d5
is defined
via
C
and a right-hand convention.
