52
CURVILINEAR
COORDINATE
SYSTEMS
goes from
qi
to
qi
+
dqi
on an edge, it is set equal to
qi.
This
is a somewhat cavalier
way
to
treat
the position dependence of these factors, although it does give all the
correct results for
our
derivations.
A
more rigorous approach, which evaluates the
mean value of the scale factors on each edge,
is
presented
in
Appendix
B
in a detailed
derivation of the curvilinear curl.
Following
this
approach, the differential element’s volume is simply
(3.23)
where the
hi’s
are all evaluated at the point
(ql,
qz,
q3).
The differential surface
of
the
bottom shaded side
is
d*baom
=
-dqldqM~q31
,
(3.24)
where
the
minus sign occurs because
the
surface normal
is
antiparallel to
q3.
In
contrast, the differential surface of the top shaded side is
(41.42.43)
datop
=
dqldq2hlh243
(3.25)
(41.42143
+43)
The minus sign is absent because now the surface normal is parallel to
q3.
In
th~s
case,
hl,
hz,
and the basis vector
q3
are evaluated at the point
(ql,
q2,
q3
+
dq3).
3.4.3
The
Displacement
Vector
The displacement vector
di
plays a central role in the mathematics
of
curvilinear
systems. Once the
form
of
dF
is known, equations for most of the vector operations
can be easily determined. From multivariable, differential calculus,
di
can be written
(3.26)
AS
we showed in Equation
3.22,
this
can be written using the scale factors
as
dI;
=
dq.h.
1 1%.
A.
(3.27)
In a Cartesian system
qi
=
xi,
qi
=
Ci,
and
hi
=
1,
so
Equation
3.27
becomes the
familiar
dF
=
d&.
(3.28)
In cylindrical coordinates,
hl
=
h,
=
1,
hz
=
h+
=
p,
and
h3
=
h,
=
1
so
(3.29)
