Cylindrical and spherical coordinates
Problem 9.540 · easy
Convert the spherical point \( \displaystyle \left(6, \frac{\pi}{3}, \pi\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}0\\-6\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(0, \frac{\pi}{3}, -6\right) \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the conversion formulas from spherical to cylindrical coordinates. The intermediate matrix equality confirms the calculated values for r and z, and the final answer matches the stated result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the conversion formulas from spherical to cylindrical coordinates. The intermediate matrix equality confirms the calculated values for r and z, and the final answer matches the stated result.gpt-oss:20b: pass 2026-10-11gpt-oss:20b: fail (error) 2026-10-11 — The conversion formulas are misapplied: r=ρ sinφ gives 3√3, not 0; θ stays π, not π/3; z=ρ cosφ gives 3, not –6. The stated answer is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly calculate the cylindrical coordinates from the given spherical point, instead presenting a tautological matrix equality. It does not demonstrate the conversion steps (r = 6 sin(pi) = 0, theta = pi/3, z = 6 cos(pi) = -6) required to justify the answer.
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 2026-10-11 with SymPy 1.14.0.