Cylindrical and spherical coordinates
Problem 9.296 · easy
Convert the rectangular point \( \displaystyle \left(0, -4, -4\right) \) to cylindrical coordinates with \( \displaystyle r > 0 \), \( \displaystyle 0 \le \theta < 2\pi \).
- \[ 4 \]r = √(x² + y²).✓ Proved
- \[ \left[\begin{matrix}0\\-4\end{matrix}\right] \]θ = 3*pi/2 puts the point in the right quadrant.✓ Proved
Answer \( \left(4, \frac{3 \pi}{2}, -4\right) \)
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 |
| 2 | ✓ 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: fail (misleading) — The solution omits the z‑coordinate and provides no justification for choosing θ=3π/2 beyond a vague statement about the quadrant; a student could be misled into thinking the answer is complete when it is not.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate or state the value of z, which is required for cylindrical coordinates. While the final answer is correct, the derivation is incomplete because it does not show that z = -4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly calculate or state the value of z, which is required for cylindrical coordinates. While the final answer is correct, the derivation is incomplete because it does not show that z = -4.gpt-oss:20b: fail (misleading) 2026-10-05 — The solution omits the z‑coordinate and provides no justification for choosing θ=3π/2 beyond a vague statement about the quadrant; a student could be misled into thinking the answer is complete when it is not.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly calculate or justify the value of theta, instead presenting a tautological matrix equality that does not demonstrate the conversion logic. It also omits the calculation for the z-coordinate, which is trivial but part of the full conversion.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.