Cylindrical and spherical coordinates
Problem 9.458 · easy
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(4, \frac{7 \pi}{4}, 0\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}0\\0\\4\end{matrix}\right] \]x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.✓ Proved
Answer \( \left(0, 0, 4\right) \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the result is (0, 0, 4), but the formulas provided (x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ) with φ = 0 yield x = 0, y
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the result is (0, 0, 4), but the formulas provided (x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ) with φ = 0 yield x = 0, yqwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.