Page 435 - Advanced engineering mathematics

12.10 Curvilinear Coordinates 415

z

ρ P
φ

y
θ
x

FIGURE 12.28 Spherical coordinates of
Example 12.29.

EXAMPLE 12.29 Spherical Coordinates
Any point P having rectangular coordinates (x, y, z) also has unique spherical coordinates
(ρ,θ,ϕ).Here ρ is the distance from the origin to P, θ is the angle of rotation from the ori-
gintothelinefrom (0,0) to (x, y) in the x,y-plane, and ϕ is the angle of declination from the
positive z-axis to the line from the origin to P. These are indicated in Figure 12.28. Thus, ρ ≥ 0,
0 ≤ θ< 2π and 0 ≤ ϕ ≤ π.
Rectangular and spherical coordinates are related by
x = ρ cos(θ)sin(ϕ), y = ρ sin(θ)sin(ϕ), z = ρ cos(ϕ).

Again with care in using the inverse trigonometric functions, these equations can be inverted to
read

y

2
2
2
ρ = x + y + z ,θ = arcsin
2
2
x + y + z 2

y
.
ϕ = arccos
2
2
x + y + z 2
The coordinate systems of these examples may appear quite dissimilar, but they share a
common feature if we adopt a particular point of view. Let P 0 :(x 0 , y 0 , z 0 ) be a point in rectangular
coordinates. Observe that P 0 is the point of intersection of the planes x = x 0 , y = y 0 , and z = z 0 ,
which are called coordinate surfaces for rectangular coordinates.
Now suppose P 0 has cylindrical coordinates (r 0 ,θ 0 , z 0 ). Look at the corresponding coordi-
nate surfaces for these coordinates. In 3-space, the surface r = r 0 is a cylinder of radius r 0 about
the origin. The surface θ = θ 0 is a half-plane with edge on the z-axis and making an angle θ 0
with the positive x-axis. And the surface z = z 0 is the same as in rectangular coordinates, a plane
in 3-space parallel to the x, y-plane. The point P 0 : (r 0 ,θ 0 , z 0 ) is the intersection of these three
cylindrical coordinate surfaces.
Spherical coordinates can be viewed in the same way. Suppose P 0 has spherical coordinates
(ρ 0 ,θ 0 ,ϕ 0 ). The coordinate surface ρ = ρ 0 is a sphere of radius ρ 0 about the origin. The surface
θ = θ 0 is a half-plane with one edge along the z-axis, as in cylindrical coordinates. And the
surface ϕ = ϕ 0 is an infinite cone with vertex at the origin and making an angle ϕ with the z-axis.
These surfaces intersect at P 0 (Figure 12.29).

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October 14, 2010 14:53 THM/NEIL Page-415 27410_12_ch12_p367-424