Cylindrical and spherical coordinates
Problem 9.535 · easy
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(2, 0, 0\right) \) 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 \( \left(0, 0, 2\right) \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: pass — The solution correctly applies the standard conversion formulas for spherical to rectangular coordinates. For (2, 0, 0), x = 2*sin(0)*cos(0) = 0, y = 2*sin(0)*sin(0) = 0, and z = 2*cos(0) = 2, yielding (0, 0, 2).
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the standard conversion formulas for spherical to rectangular coordinates. For (2, 0, 0), x = 2*sin(0)*cos(0) = 0, y = 2*sin(0)*sin(0) = 0, and z = 2*cos(0) = 2, yielding (0, 0, 2).gpt-oss:20b: pass 2026-10-11gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the standard conversion formulas for spherical to rectangular coordinates, yielding the correct result.
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.