Cylindrical and spherical coordinates
Problem 9.300 · easy
Convert the spherical point \( \displaystyle \left(6, \frac{\pi}{6}, \frac{\pi}{6}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}3\\3 \sqrt{3}\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(3, \frac{\pi}{6}, 3 \sqrt{3}\right) \)
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 conversion formulas used (r = ρ sin φ, θ = θ, z = ρ cos φ) as a sentence, relying instead on a comment in an equation line. More critically, it does not verify that the input spherical coordinates are valid or explain the substitution steps, making the derivation opaque. However, the primary failure is that the 'sentence' part of the solution is missing; the provided line is just an equation check. A proper solution must state the method.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the conversion formulas used (r = ρ sin φ, θ = θ, z = ρ cos φ) as a sentence, relying instead on a comment in an equation line. More critically, it does not verify that the input spherical coordinates are valid or explain the substitution steps, making the derivation opaque. However, the primary failure is that the 'sentence' part of the solution is missing; the provided line is just an equation check. A proper solution must state the method.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution is incomplete and confusing. It presents a matrix equality that does not clearly show the conversion steps or the final cylindrical coordinates $(r, \theta, z)$. Specifically, it fails to explicitly state the values for $r$, $\theta$, and $z$ derived from the given spherical coordinates $(\rho, \theta, \phi) = (6, \pi/6, \pi/6)$. While the final answer provided in the problem statement is correct, the solution text itself is not a valid derivation.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.