Questions: Spherical Coordinates

Question 1

A point has spherical coordinates (ρ, φ, θ) = (4, π/2, π/3). Which description correctly locates this point?

• A. On the positive z-axis, 4 units from the origin
• B. In the xy-plane, 4 units from the origin, at 60° from the positive x-axis
• C. At height z = 4cos(π/3) = 2, rotating 60° around the z-axis
• D. At the south pole, distance 4 from the origin

Answer: (B)

φ = π/2 means the polar angle from the z-axis is 90°, placing the point in the equatorial xy-plane. From there, ρ = 4 gives the distance from the origin, and θ = π/3 = 60° is the azimuthal angle in the xy-plane. So x = 4sin(π/2)cos(π/3) = 4·1·½ = 2, y = 4sin(π/2)sin(π/3) = 4·(√3/2) = 2√3, z = 4cos(π/2) = 0. The point lies in the xy-plane. The most common error is thinking φ = π/2 means 'halfway up' — it actually means exactly on the equatorial plane, since φ is measured from the z-axis.

Question 2

A student computes the volume of a sphere of radius R in spherical coordinates by integrating ∫₀²π ∫₀π ∫₀ᴿ dρ dφ dθ and gets 2π²R. The correct answer is (4/3)πR³. What did the student do wrong?

• A. The φ limits should be 0 to 2π, not 0 to π
• B. The volume element is ρ² sinφ dρ dφ dθ — the student integrated without the Jacobian factors
• C. The θ limits should be 0 to π for a full sphere
• D. Spherical coordinates cannot be used to compute volumes — only surface areas

Answer: (B)

The volume element in spherical coordinates is dV = ρ² sinφ dρ dφ dθ, not dρ dφ dθ. The factor ρ² accounts for the fact that a thin shell at radius ρ has area 4πρ², not 4π. The factor sinφ accounts for the fact that circles of latitude shrink near the poles — at φ = 0 or π, the 'ring' in the θ-direction has zero circumference. Omitting these factors ignores the geometry of the coordinate system and gives a dimensionally incorrect result. The Jacobian is never optional when changing coordinate systems.

Question 3

In the mathematics convention for spherical coordinates, φ = 0 corresponds to the equatorial plane (the xy-plane).

• T. True
• F. False

Answer: False

φ = 0 corresponds to the positive z-axis — the 'north pole' direction. As φ increases from 0 to π, the angle sweeps from the north pole down through the equatorial plane (φ = π/2) to the south pole (φ = π). This is the mathematics convention. In physics, the roles of φ and θ are often reversed, which is a persistent source of confusion. The key rule to remember: in mathematics, φ is the polar angle from the z-axis, ranging from 0 to π.

Question 4

In spherical coordinates, allowing φ to range from 0 to 2π instead of 0 to π would cause every interior point of the sphere (except those on the z-axis) to be described by exactly two different coordinate triples.

• T. True
• F. False

Answer: True

If φ ∈ [0, 2π], the angles φ and 2π − φ both describe points on the same circle of latitude — one above the equatorial plane, one below. For any interior point not on the z-axis, there are two values of φ in [0, 2π] that place you at the same (x, y, z) location (with corresponding adjustments to θ). This is why the range φ ∈ [0, π] is the correct convention: it gives a unique coordinate representation for every point (except the z-axis, where θ is undefined). Using φ ∈ [0, 2π] would double-count the entire sphere.

Question 5

Why does the volume element in spherical coordinates contain the factor sinφ, and what would go wrong if you omitted it from a triple integral?

Model answer: The sinφ factor corrects for the fact that circles of latitude at polar angle φ have circumference 2πρ sinφ — shrinking to zero at the poles (φ = 0 and π) and reaching maximum at the equator (φ = π/2). When you integrate over φ, each thin 'ring' in the θ-direction contributes area proportional to its circumference, which is ρ sinφ dθ. Without sinφ, every ring would be treated as having the same size regardless of latitude, massively over-counting the polar regions. Integrating the sphere's volume without sinφ gives 2π²R instead of (4/3)πR³.

The geometric picture is: a small volume element in spherical coordinates is approximately a rectangular box with side lengths dρ (radial), ρ dφ (arc in the φ-direction), and ρ sinφ dθ (arc in the θ-direction). The product of these three lengths is ρ² sinφ dρ dφ dθ. The factor ρ² arises because arcs in the angular directions grow with distance from the origin; sinφ arises because the θ-arc shrinks near the poles. Both factors are essential and both come directly from the geometry of the coordinate system.