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Cylindrical and spherical coordinates

Problem 9.378 · easy

Convert the rectangular point \( \displaystyle \left(4, 0, 0\right) \) to spherical coordinates (\( \displaystyle \rho > 0 \), \( \displaystyle 0 \le \theta < 2\pi \), \( \displaystyle 0 \le \varphi \le \pi \)).

\[ 4 \]
ρ = √(x² + y² + z²).✓ Proved
\[ 0 \]
cos φ = z/ρ, so φ = pi/2.✓ Proved
\[ \left[\begin{matrix}4\\0\end{matrix}\right] \]
θ = 0 matches x and y.✓ Proved

Answer \( \left(4, 0, \frac{\pi}{2}\right) \)

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
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly justify why theta is 0 rather than pi, despite y=0. While the final answer is correct, the reasoning 'theta = 0 matches x and y' is insufficient because x=4 > 0 is the critical condition that selects theta=0 over theta=pi. A rigorous solution must check the sign of x (or use atan2) to determine the correct quadrant for theta.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly justify why theta is 0 rather than pi, despite y=0. While the final answer is correct, the reasoning 'theta = 0 matches x and y' is insufficient because x=4 > 0 is the critical condition that selects theta=0 over theta=pi. A rigorous solution must check the sign of x (or use atan2) to determine the correct quadrant for theta.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly calculate or justify the value of theta, merely asserting it matches x and y without showing the arctan logic or handling the quadrant correctly for the point (4,0). Additionally, the intermediate steps are presented as trivial identities rather than derivations, making the reasoning opaque.
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.