Cylindrical and spherical coordinates
Problem 9.377 · easy
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(2, \frac{\pi}{6}, 0\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}0\\0\\2\end{matrix}\right] \]x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.✓ Proved
Answer \( \left(0, 0, 2\right) \)
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution uses the formula z = ρ cos φ, which corresponds to φ being the angle from the z-axis (physics convention). However, the problem notation (ρ, θ, φ) typically implies the mathematics convention where φ is the azimuthal angle and θ is the polar angle, or vice versa depending on the text, but standard calculus texts (e.g., Stewart) use (ρ, θ, φ) with φ as the angle from the positive z-axis. Wait, let's look closer. In standard calculus (Stewart), x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, where φ is the angle from the z-axis. If φ=0, the point is on the z-axis. The calculation yields (0,0,2). This is actually correct for the standard calculus convention. Why did I think it was wrong? Let me re-read the prompt's implied convention. Usually, if the order is (ρ, θ, φ), θ is azimuthal and φ is polar. If the order is (ρ, φ, θ), it might be different. The solution explicitly states the formulas used. If the formulas are consistent with the standard definition of the symbols in the problem statement, it is correct. However, many contexts use (ρ, θ, φ) where θ is polar and φ is azimuthal (ISO convention). If the problem implies ISO convention, the answer is wrong. But without external context, we must judge the internal consistency. The solution states the formulas. Are the formulas correct for the symbols? In standard US calculus, φ is polar. So (2, π/6, 0) -> ρ=2, θ=π/6, φ=0. z = 2 cos(0) = 2. x = 2 sin(0) cos(π/6) = 0. y = 0. Result (0,0,2). This is correct for the standard convention. Is there an error? The prompt asks to review adversarially. Is there a subtle error? The input is (ρ, θ, φ) = (2, π/6, 0). The solution uses x = ρ sin φ cos θ. This assumes φ is the polar angle. If the problem intended φ to be the azimuthal angle (common in physics/ISO), the answer would be different. However, in the absence of specified convention, the standard calculus convention (φ=polar) is the default assumption for such problems in US textbooks. The solution is internally consistent and follows the standard convention. Wait, look at the equation line. It says 'unchecked'. The algebra is trivial. The sentence is just the formula. Is the formula correct? Yes, for standard calculus. Is the application correct? Yes. So why fail? Maybe I should pass? Let me double check if there's a trick. (2, π/6, 0). ρ=2. θ=π/6. φ=0. If φ is polar, it's on the z-axis. Correct. If φ is azimuthal, it's on the x-axis (since θ would be polar? No, usually θ is azimuthal in math). If the problem uses ISO (r, θ, φ) where θ is polar and φ is azimuthal, then x = r sin θ cos φ = 2 sin(π/6) cos(0) = 1. y = 2 sin(π/6) sin(0) = 0. z = 2 cos(π/6) = √3. The answer (0,0,2) is definitely for the Math convention. Is the Math convention the standard for the notation (ρ, θ, φ)? Yes, Stewart uses (ρ, θ, φ) with φ as polar. So the solution is correct. I will pass.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.