46
CURVILINEAR COORDINATE SYSTEMS
x
=
pcos4
y
=
psin+
(3.5)
z=z
and the corresponding inverse equations
p
=
dw
4
=
tan-'
(Yh)
(3.6)
z=z
govern the relationship between cylindrrcal coordinates and the coordinates
of
a
superimposed Cartesian system,
as
shown in Figure 3.2(a).
The unit basis vectors for the cylindrical system are shown in Figure 3.2(b). Each
basis vector points in the direction that
P
moves when the corresponding coordinate
is
increased. For example, the direction
of
SP
is
found
by watching how
P
moves as
p
is increased.
This
method can
be
used to determine the directions of basis vectors
for any set
of
coordinates. Unlike the Cartesian system, the cylindrical basis vectors
are not fixed.
As
the point
P
moves, the directions
of
i$
and
i?+
both change.
Also
notice that if
P
lies exactly on the
z-axis,
i.e.,
p
=
0,
the directions
of
Sp
and
6,
are
undefined.
The cylindrical coordinates, taken in the order
(p,
4,
z),
form
a right-handed
sys-
tem.
If
you align your right hand along
GP,
and then curl your fingers to point in the
direction
of
i?+,
your thumb will point in the
GZ
direction. The basis vectors are also
orthononnal since
6,
.
6,
=
6,
.
Q
=
Q
.
e+
=
0
e,
.
*p
=
e,
.
c,
=
c,
.
Q
=
1.
The position vector expressed in cylindrical coordinates is
r"
Ap
Y
X
z
6,
X
(4
(b)
Figure
3.2
The
Cylindrical System
(3.7)
(3.8)
