Cylindrical and spherical coordinates
Problem 9.214 · easy
Convert the spherical point \( \displaystyle \left(1, \frac{3 \pi}{4}, \frac{\pi}{6}\right) \) to cylindrical coordinates.
- \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] \]r = ρ sin φ, θ = θ, z = ρ cos φ.✓ Proved
Answer \( \left(\frac{1}{2}, \frac{3 \pi}{4}, \frac{\sqrt{3}}{2}\right) \)
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 claims to convert to cylindrical coordinates but only computes the radial component $r$ and $z$, omitting the angle $ heta$ in the final vector representation. Furthermore, the equation shown is a tautology that does not demonstrate the conversion process or the value of $ heta$.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims to convert to cylindrical coordinates but only computes the radial component $r$ and $z$, omitting the angle $ heta$ in the final vector representation. Furthermore, the equation shown is a tautology that does not demonstrate the conversion process or the value of $ heta$.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the conversion formulas used, making the jump from spherical to cylindrical coordinates opaque. Additionally, the provided equation is a tautology that does not demonstrate the calculation steps (e.g., showing r = 1 * sin(π/6) = 1/2).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.