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

170 Chapter 4 Coordinate systems
We now develop the equation of the helix in cylindrical polar coordinates. By
comparing
x=3cos2t,y=3sin2t with x=rcosθ,y=rsinθ
(see Equations (4.1) and (4.2)), we haver = 3 and θ = 2t. Note thatr = 3 is the
equation of a circle, radius 3,centre the origin.
So, the projection of the helix onto thex--y plane is a circle of radius 3. Because
z(t) =t, the value ofzincreases as the parametert increases and the helix is traced
out.
Wecannowstateanalternativedefinitionofthehelixintermsofcylindricalpolar
coordinates:
r(t) =3, θ(t) =2t, z(t) =t
AsintheCartesiancase,specifyingavalueoftheparametert enablesustocalculate
particular values ofr, θ andzcorresponding to a point on the helix. This provides a
more elegant definition of the helix than that available using Cartesian coordinates.
Many problems require the use of these alternative coordinate systems in order to
simplifyanalysis.
EXERCISES4.6
1 Expressthe followingCartesian coordinates as
cylindrical polar coordinates.
(a) (−2,−1,4) (b) (0,3,−1) (c) (−4,5,0)
2 Expressthe followingcylindrical polar coordinates as
Cartesian coordinates.
(a) (3,70 ◦ ,7) (b) (1,200 ◦ ,6) (c) (5,180 ◦ ,0)
3 Describe the surface defined by
(a) z=0
(b) z=−1
(c) r=2,z=1
(d) θ =90 ◦ ,z=3
(e) r=2,0z4
Solutions
1 (a) ( √ 5,206.57 ◦ ,4)
(b) (3,90 ◦ ,−1)
(c) ( √ 41,128.66 ◦ ,0)
2 (a) (1.0261,2.8191,7)
(b) (−0.9397,−0.3420,6)
(c) (−5,0,0)
3 (a) thex--y plane
(b) aplaneparallelto thex--yplaneand1unit
below it
(c) acircle,radius2,parallelto thex--yplaneand
with centreat(0,0,1)
(d) aline 3 units above the positiveyaxisand
parallelto it
(e) thecurvedsurfaceofacylinder,radius2,height4
4.7 SPHERICALPOLARCOORDINATES
When problems involve spheres, for example modelling the flow of oil around a ball
bearing,itmaybeusefultousesphericalpolarcoordinates.Thepositionofapointis
given bythreecoordinates, (R,θ,φ). These areillustratedinFigure 4.28.