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

166 Chapter 4 Coordinate systems
Solutions
1 This isaquarter circle ofradius2asshown in
FigureS.8.
2 FigureS.9 illustratesthe surface. OPis setat30 ◦ to
thexaxis;OQis at45 ◦ to thexaxis. OP = OQ = 2.
y
y
Q
2
P
O
x
O
x
FigureS.8
FigureS.9
4.6 CYLINDRICALPOLARCOORDINATES
Consider the problem of studying the flow of water around a cylinder. A problem like
this would be studied by engineers when investigating the forces exerted by the sea on
the cylindrical supports of oil-rigs. It is often mathematically convenient to choose a
coordinate system that fits the shape of the object being described. It makes sense here
toselect a cylindricalcoordinate system.
Cylindrical polar coordinates comprise polar coordinates with the addition of a
vertical, or z, axis. Figure 4.21 illustrates a typical point, P, and its cylindrical polar
coordinates.
The point Q is in the x--y plane and lies directly below P. Q is the projection of P
onto thex--y plane.
Consider a point P in three-dimensional space, with Cartesian coordinates (x,y,z).
We can also describe the position of P using cylindrical polar coordinates. To do this,
the x and y coordinates are expressed as their equivalent polar coordinates, while the
z coordinate remains unaltered. Hence the cylindrical polar coordinates of a point have
theform (r,θ,z).
Recall thatris the length of the arm OQ (see Figure 4.21); that is, it is the distance
of a pointinthex--yplane from the origin, and sor 0.The angle θ is measured from
thepositivexaxistothearmOQandso θ hasvaluesbetween0 ◦ and360 ◦ or2πradians.
Finally, z is positive for points above the x--y plane and negative for points below the
x--yplane.Insummary
r0, 0θ<2π, −∞<z<∞
z
P
O
u
z
y
x
r
Q
Figure4.21
Thecylindrical polar coordinates ofPare (r,θ,z).