Cylindrical and spherical coordinates
Problem 9.459 · easy
Convert the spherical point \( \displaystyle \left(5, \frac{\pi}{4}, 0\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}0\\5\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(0, \frac{\pi}{4}, 5\right) \)
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. A reviewer's objection to the explanation stands; see 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 (5,π/4,0) has r=5 sin(π/4)=5/√2, θ=0, z=5 cos(π/4)=5/√2. The solution incorrectly sets r=0 and z=5, giving a wrong cylindrical coordinate. The setup and calculation are incorrect.qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The spherical point (5,π/4,0) has r=5 sin(π/4)=5/√2, θ=0, z=5 cos(π/4)=5/√2. The solution incorrectly sets r=0 and z=5, giving a wrong cylindrical coordinate. The setup and calculation are incorrect.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-09 — The conversion from spherical to cylindrical coordinates is incorrect: r should be ρ sinφ=5/√2, θ=0, z=ρ cosφ=5/√2, not (0, π/4, 5).qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims the result is (0, pi/4, 5), but the calculation r = 5 sin(pi/4) yields 5/sqrt(2), not 0. The provided matrix equation is nonsensical and does not represent the conversion correctly.
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 2026-10-09 with SymPy 1.14.0.