Cylindrical and spherical coordinates
Problem 9.291 · easy
Convert the spherical point \( \displaystyle (\rho, \theta, \varphi) = \left(5, \frac{\pi}{3}, \frac{\pi}{3}\right) \) to rectangular coordinates.
- \[ \left[\begin{matrix}\frac{5 \sqrt{3}}{4}\\\frac{15}{4}\\\frac{5}{2}\end{matrix}\right] \]x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.✓ Proved
Answer \( \left(\frac{5 \sqrt{3}}{4}, \frac{15}{4}, \frac{5}{2}\right) \)
✓ 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 and arrives at the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the standard conversion formulas for spherical to rectangular coordinates and arrives at the correct result.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the standard conversion formulas for spherical coordinates (assuming the physics convention where φ is the polar angle) and the algebraic evaluation is correct.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.