Cylindrical and spherical coordinates
Problem 9.537 · easy
Convert the spherical point \( \displaystyle \left(1, \frac{7 \pi}{4}, \frac{\pi}{4}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(\frac{\sqrt{2}}{2}, \frac{7 \pi}{4}, \frac{\sqrt{2}}{2}\right) \)
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 conversion formulas or the input values, making the derivation opaque. Furthermore, the provided equation is a tautology that does not demonstrate the calculation steps (e.g., calculating r = 1 * sin(pi/4) and z = 1 * cos(pi/4)).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly state the conversion formulas or the input values, making the derivation opaque. Furthermore, the provided equation is a tautology that does not demonstrate the calculation steps (e.g., calculating r = 1 * sin(pi/4) and z = 1 * cos(pi/4)).gpt-oss:20b: pass 2026-10-11gpt-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 calculation steps (such as calculating r and z from ρ and φ), making it impossible to verify the logic or learn the method. The provided equation is a tautology that does not demonstrate the conversion process.
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.