Cylindrical and spherical coordinates
Problem 9.534 · easy
Convert the spherical point \( \displaystyle \left(2, \pi, \pi\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}0\\-2\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(0, \pi, -2\right) \)
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: passqwen3.6:27b-mlx: fail (error) — The solution provides a matrix equality that does not represent the conversion to cylindrical coordinates. It fails to explicitly state the cylindrical coordinates (r, θ, z) and the matrix shown is 2x1, missing the z-component or misrepresenting the output format entirely.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution provides a matrix equality that does not represent the conversion to cylindrical coordinates. It fails to explicitly state the cylindrical coordinates (r, θ, z) and the matrix shown is 2x1, missing the z-component or misrepresenting the output format entirely.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the conversion formulas or the intermediate steps (calculating r, theta, z) required to justify the result. It presents a tautological matrix equality that does not demonstrate the conversion process from spherical to cylindrical coordinates.gpt-oss:20b: pass 2026-10-11
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.