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

4.6 Cylindrical polar coordinates 167
We can relate the Cartesian coordinates, (x,y,z), to the cylindrical polar coordinates,
(r,θ,z). The following key point does this.
x=rcosθ r0
y=rsinθ 0θ <2π
z =z
Engineeringapplication4.4
Fluidflowalongapipe
Cylindrical polar coordinates provide a convenient framework for analysing liquid
flow down a pipe. The radial symmetry of a pipe makes it the natural choice. The
distance along the pipe is defined usingz. In order to utilize the radial symmetry of
thepipeitisnecessarytoalignthezaxiswiththecentreaxisofthepipe.Figure4.22
illustratesthearrangement.Distancefromthecentreofthepipeisdefinedbyr.The
angle θ isusedinconjunctionwithzandrtofixthepositionwithinthepipe.Atypical
problemthatmaybeanalysedisthevariationinfluidvelocitywithdistancefromthe
centreofthepipe.Forsmoothflow,liquidtendstotravelfasteratthecentreofapipe
than itdoes near the edge.
r
u
z
Figure4.22
Fluid flowalong apipe.
Pipes with metal walls are often used to guide electromagnetic waves, rather
than fluids, in high-powered microwave communications systems. They are termed
waveguides.Mathematicallyanalysingthewaveguide’spropagationmodesismade
much simplerby using cylindrical polar coordinates.
Example4.9 The Cartesian coordinates of P are (4,7,−6). State the cylindrical polar coordinates
ofP.
Solution We have
x=4, y=7, z=−6
Usingx = 4 andy = 7 the values ofrand θ are found to ber = 8.0623,θ = 60.26 ◦
(see Example 4.6). The z coordinate remains unchanged. Hence the cylindrical polar
coordinates are (8.0623,60.26 ◦ ,−6).