Cylindrical and spherical coordinates
Problem 9.212 · easy
Convert the spherical point \( \displaystyle \left(5, \frac{7 \pi}{4}, \frac{3 \pi}{4}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}\frac{5 \sqrt{2}}{2}\\- \frac{5 \sqrt{2}}{2}\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(\frac{5 \sqrt{2}}{2}, \frac{7 \pi}{4}, - \frac{5 \sqrt{2}}{2}\right) \)
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 provides no text explaining the conversion formulas or the calculation steps, making it impossible to verify the logic or teach the method. Additionally, the single matrix equation shown is an identity that does not demonstrate the derivation of the cylindrical coordinates from the spherical ones.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution provides no text explaining the conversion formulas or the calculation steps, making it impossible to verify the logic or teach the method. Additionally, the single matrix equation shown is an identity that does not demonstrate the derivation of the cylindrical coordinates from the spherical ones.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution provides no derivation or explanation for the conversion formulas used, merely stating them alongside an identity check. It fails to explicitly calculate the intermediate values (x, y, z) or show how the cylindrical coordinates (r, theta, z) are derived from the spherical coordinates (rho, theta, phi), making the solution incomplete and uninformative.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.