Cylindrical and spherical coordinates
Problem 9.294 · easy
Convert the spherical point \( \displaystyle \left(4, \frac{\pi}{6}, \frac{2 \pi}{3}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}2 \sqrt{3}\\-2\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(2 \sqrt{3}, \frac{\pi}{6}, -2\right) \)
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 fails to explicitly state the cylindrical coordinates as a triple $(r, \theta, z)$. While the matrix contains the correct $r$ and $z$ values, it omits $\theta = \pi/6$, which is a required component of the answer. Additionally, the justification text is embedded in the equation line rather than being a distinct logical step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the cylindrical coordinates as a triple $(r, \theta, z)$. While the matrix contains the correct $r$ and $z$ values, it omits $\theta = \pi/6$, which is a required component of the answer. Additionally, the justification text is embedded in the equation line rather than being a distinct logical step.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly calculate or state the cylindrical radius r, providing only a 2x1 matrix for z and an implicit r. Furthermore, the conversion formulas are stated but not applied in a clear, step-by-step manner to derive the final vector, making the logic opaque and incomplete.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.