082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

168 Chapter 4 Coordinate systems
Example4.10 Describe the figure defined by
(a) 1r2,θ=60 ◦ ,−1z1
(b)r=1,0 ◦ θ90 ◦ ,0z2
z
A
B
y
P
Q
x
P
C
Q
y
O
60°
x
D
Figure4.23
AtP,r=1;atQ,r=2.
Figure4.24
On the line AB,z = 1; onthe line CD,z = −1.
Solution (a) Consider therand θ coordinates first. Thercoordinate varies from 1 to 2 while θ
is fixed at 60 ◦ . This represents the line PQ as shown in Figure 4.23. At P the value
ofris 1; at Q the value ofris 2. The length of PQ is 1 and it is inclined at 60 ◦ to
thexaxis.
Now, we note thatzvaries from −1 to 1. We imagine the line PQ moving in the
z direction fromz = −1 toz = 1. This movement sweeps out a plane. Figure 4.24
illustratesthis.
(b) The r coordinate is fixed at r = 1. The θ coordinate varies from 0 ◦ to 90 ◦ . This
produces the quarter circle, AB, as shown in Figure 4.25. At A,r = 1, θ = 0 ◦ ; at
B,r=1,θ =90 ◦ .
z
D
y
C
B
B
A
y
O
A
x
x
Figure4.25
As θ varies from0 ◦ to 90 ◦ ,aquarter
circle is sweptout.
Figure4.26
Aszvaries from0to 2,the curveAB
sweepsoutthe curved surface.