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Cylindrical and spherical coordinates

Problem 9.215 · easy

Convert the spherical point $ \displaystyle \left(6, \frac{\pi}{4}, \frac{\pi}{4}\right) $ to cylindrical coordinates.

\[ \left[\begin{matrix}3 \sqrt{2}\\3 \sqrt{2}\end{matrix}\right] \]
r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved

Answer $ \left(3 \sqrt{2}, \frac{\pi}{4}, 3 \sqrt{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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to compute the cylindrical radius $r = \rho \sin \phi$ and height $z = \rho \cos \phi$ explicitly, instead presenting a tautological matrix equality that does not demonstrate the conversion steps or justify the numerical values.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the cylindrical radius $r = \rho \sin \phi$ and height $z = \rho \cos \phi$ explicitly, instead presenting a tautological matrix equality that does not demonstrate the conversion steps or justify the numerical values.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to compute the cylindrical coordinates explicitly. It presents an identity matrix equality that does not correspond to the conversion formulas or the given input values, and it omits the calculation of the z-coordinate entirely.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.