Cylindrical and spherical coordinates
Problem 9.376 · easy
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(2, \frac{3 \pi}{2}, \pi\right) \) 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 from spherical to rectangular coordinates. The calculation yields the correct result (0, 0, -2), and the equation is verified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the standard conversion formulas from spherical to rectangular coordinates. The calculation yields the correct result (0, 0, -2), and the equation is verified.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the standard spherical to rectangular conversion formulas and arrives at the correct coordinates.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.