Cylindrical and spherical coordinates
Problem 9.213 · easy
Convert the spherical point \( \displaystyle \left(3, 0, \frac{\pi}{6}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}\frac{3}{2}\\\frac{3 \sqrt{3}}{2}\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(\frac{3}{2}, 0, \frac{3 \sqrt{3}}{2}\right) \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | converted back to rectangular in floating point |
Reviewers
gpt-oss:20b: fail (error) — The spherical point (3,0,π/6) has φ=0, so r=ρ sinφ=0 and z=ρ cosφ=3. The solution incorrectly uses φ=π/6, yielding r=3/2 and z=3√3/2, and also mis‑assigns the θ coordinate. The conversion is wrong.qwen3.6:27b-mlx: fail (error) — The solution uses the conversion formula z = ρ cos φ, which corresponds to the convention where φ is the polar angle (angle from the positive z-axis). However, the standard mathematical convention (used in most calculus texts) defines φ as the azimuthal angle in the xy-plane and θ as the polar angle. Under the standard convention, the given point (3, 0, π/6) would imply ρ=3, θ=0, φ=π/6, leading to z = 3 sin(π/6) = 3/2, not 3√3/2. Even if the non-standard convention is assumed, the solution fails to explicitly state this convention, leading to ambiguity and potential error for a student expecting the standard definition. Furthermore, the output is presented as a 2x1 matrix, which is not the standard format for cylindrical coordinates (r, θ, z).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution uses the conversion formula z = ρ cos φ, which corresponds to the convention where φ is the polar angle (angle from the positive z-axis). However, the standard mathematical convention (used in most calculus texts) defines φ as the azimuthal angle in the xy-plane and θ as the polar angle. Under the standard convention, the given point (3, 0, π/6) would imply ρ=3, θ=0, φ=π/6, leading to z = 3 sin(π/6) = 3/2, not 3√3/2. Even if the non-standard convention is assumed, the solution fails to explicitly state this convention, leading to ambiguity and potential error for a student expecting the standard definition. Furthermore, the output is presented as a 2x1 matrix, which is not the standard format for cylindrical coordinates (r, θ, z).gpt-oss:20b: fail (error) 2026-10-04 — The spherical point (3,0,π/6) has φ=0, so r=ρ sinφ=0 and z=ρ cosφ=3. The solution incorrectly uses φ=π/6, yielding r=3/2 and z=3√3/2, and also mis‑assigns the θ coordinate. The conversion is wrong.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly identifies the cylindrical coordinates as (r, z) instead of (r, θ, z). It omits the theta component entirely and presents a 2D vector, whereas cylindrical coordinates require three components.gpt-oss:20b: pass 2026-10-04
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/cylindrical_spherical_coords, checked 2026-10-04 with SymPy 1.14.0.