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

4.6 Cylindrical polar coordinates 169
Examining the z coordinate, we see thatzvaries from 0 to 2. As z varies from
0 to 2, we imagine the curve AB sweeping out the curved surface as shown in Figure4.26.AtC,r
= 1, θ = 0 ◦ ,z = 2;atD,r = 1, θ = 90 ◦ ,z = 2.Thissurfaceis
partof a cylinder.
Iftherangeofvaluesofacoordinateisnotgivenitisunderstoodthatthatvariable
variesacrossallitspossiblevalues.Forexample,acurvemaybedescribedbyr = 1,
z = −2.Herethereisnomentionofthevaluesthat θ canhave.Itisassumedthat θ
can have its fullrange of values, thatis0 ◦ to360 ◦ .
Engineeringapplication4.5
Helicalantennas
The helix is a shape commonly found inengineering. For example, the springs used
inacar’ssuspensionoftenhave ahelicalshape.Helicalantennaswereinvented by
John Kraus in the 1940s and since then have been used extensively in a variety of
applications including space exploration, satellite communications and mobile telephony.
Developing a mathematical definition of a helix is essential to analysing its
electromagnetic properties.
Wecansetupacylindricalpolarcoordinatesystemwiththezaxisalignedwiththe
axis of the helix as shown in Figure 4.27. If we were to look at the helix along the
directionofthezaxis,allwewouldseewouldbeacircle.Wesaythattheprojectionof
the helix onto thex--y plane isacircle.
z
y
x
Figure4.27
Helix along thezaxis.
Suppose a particular helix can be defined parametrically by the Cartesian equations
x(t) =3cos2t, y(t) =3sin2t, z(t) =t
wheret is varied over a particular range in order to generate the finite length helix
required. By specifying a particular value oft these equations enable us to calculate
particular values ofx,yandzcorresponding toapoint on the helix.
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