Multivariate Calculus - Bruce E. Shapiro

LECTURE 7. CYLINDRICAL AND SPHERICAL COORDINATES 49
Spherical Coordinates
In spherical coordinates each point P = (x, y, z) in Cartesian coordinates is
represented by a triple (ρ, θ, φ) where:
ρ (the Greek letter “rho”) is the distance from the origin to P.
θ (the Greek letter “theta”) is the same as in cylindrical and polar coordinates.
φ (the Greek letter “phi’)’ is the angle between the z axis and the line from the
origin to P.
Given the spherical coordinates (ρ, θ, φ), to find the Cartesian coordinates (x, y, z),
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z = ρ cos φ
Given the Cartesian coordinates (x, y, z), to find the spherical coordinates (ρ, θ, φ),
ρ 2 = x 2 + y 2 + z 2
tan θ = y/x
cos φ = z/ √ x 2 + y 2 + z 2
Figure 7.2: Spherical coordinates.
Example 7.4 Convert the equation 2x 2 +2y 2 −4z 2 = 0 from Cartesian Coordinates
to Spherical coordinates.
Math 250, Fall 2006 Revised December 6, 2006.