Calculus 2nd Edition Rogawski

722 C H A P T E R 13 VECTOR GEOMETRY
Spherical Coordinates
ρ = distance from origin
θ = polar angle in the xy-plane
φ = angle of declination from the vertical
In some textbooks, θ is referred to as the
azimuthal angle and φ as the polar angle.
The radial coordinate r of Q = (x, y, 0) is r = ρ sin φ, and therefore,
x = r cos θ = ρ cos θ sin φ, y = r sin θ = ρ sin θ sin φ, z = ρ cos φ
Spherical to rectangular Rectangular to spherical
x = ρ cos θ sin φ ρ =
√
x 2 + y 2 + z 2
y = ρ sin θ sin φ
tan θ = y x
z = ρ cos φ
cos φ = z ρ
z
z = ρ cos φ = 3 cos π √
2
4 = 3 2 = 3√ 2
2
( √ )
Now consider the projection Q = (x, y, 0) = 3 2
4 , 3√ 6
4 , 0 (Figure 8). The radial coorφ
=
π
4
ρ = 3
P
EXAMPLE 4 From Spherical to Rectangular Coordinates Find the rectangular coordinates
of P = (ρ, θ, φ) = ( 3, π 3 , π 4
)
, and find the radial coordinate r of its projection Q
onto the xy-plane.
Solution By the formulas above,
θ = π dinate r of Q satisfies
3
r
y
(
x
Q
FIGURE
(
8 Point with spherical coordinates
r 2 = x 2 + y 2 =
3,
π
3 , π )
4 .
P = (2, −23, 3)
5
φ
z
3
x = ρ cos θ sin φ = 3 cos π 3 sin π ( ) √
1 2
4 = 3 2 2 = 3√ 2
4
(√ )
y = ρ sin θ sin φ = 3 sin π 3 sin π √
3 2
4 = 3 2 2 = 3√ 6
4
Therefore, r = 3/ √ 2.
3 √ ) 2 (
2
+
4
3 √ ) 2
6
= 9 4 2
EXAMPLE 5 From Rectangular to Spherical Coordinates Find the spherical coordinates
of the point P = (x, y, z) = (2, −2 √ 3, 3).
√
Solution The radial coordinate is ρ = 2 2 + (−2 √ 3) 2 + 3 2 = √ 25 = 5. The angular
coordinate θ satisfies
tan θ = y x = −2√ 3
= − √ 3 ⇒ θ = 2π 2
3 or 5π y
3
θ = 5π 3
Since the point (x, y) = (2, −2 √ 3) lies in the fourth quadrant, the correct choice is θ = 5π x
3
(Figure 9). Finally, cos φ =
ρ z = 3 5 and so φ = cos−1 5 3 ≈ 0.93. Therefore, P has spherical
FIGURE 9 Point with rectangular
coordinates (2, −2 √ 3, 3). coordinates ( 5, 5π 3 , 0.93) .
Figure 10 shows the three types of level surfaces in spherical coordinates. Notice that
if φ ̸= 0, π 2 or π, then the level surface φ = φ 0 is the right circular cone consisting of
points P such that OP makes an angle φ 0 with the z-axis. There are three exceptional
cases: φ = π 2
defines the xy-plane, φ = 0 is the positive z-axis, and φ = π is the negative
z-axis.