Cylindrical and spherical coordinates
Problem 9.371 · easy
Convert the rectangular point \( \displaystyle \left(2 \sqrt{3}, 2, 0\right) \) 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
- \[ \left[\begin{matrix}2 \sqrt{3}\\2\end{matrix}\right] \]θ = pi/6 matches x and y.✓ Proved
Answer \( \left(4, \frac{\pi}{6}, \frac{\pi}{2}\right) \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the spherical coordinates. Although the presentation is minimalistic, the equations provided verify the values of ρ, φ, and θ against the given rectangular coordinates without error.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the spherical coordinates. Although the presentation is minimalistic, the equations provided verify the values of ρ, φ, and θ against the given rectangular coordinates without error.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly assigns the angle pi/6 to theta, whereas tan(theta) = y/x = 1/sqrt(3) implies theta = pi/6 is actually correct for the point (2sqrt(3), 2). Wait, let me re-evaluate. x = 2sqrt(3), y = 2. tan(theta) = 2 / (2sqrt(3)) = 1/sqrt(3). This corresponds to theta = pi/6. The stated answer is (4, pi/6, pi/2). The solution says theta = pi/6 matches x and y. This is correct. Let me re-read the prompt's stated answer. Stated answer: (4, pi/6, pi/2). My calculation: rho = 4. phi = pi/2. theta = pi/6. The solution is correct. Why did I think it was wrong? Ah, I need to check if the order of spherical coordinates is standard. Usually (rho, theta, phi) or (rho, phi, theta). The problem specifies (rho > 0, 0 <= theta < 2pi, 0 <= phi <= pi). This implies the order is (rho, theta, phi) or just lists the variables. The stated answer is (4, pi/6, pi/2). If the order is (rho, theta, phi), then theta=pi/6, phi=pi/2. This matches. If the order is (rho, phi, theta), then phi=pi/6, theta=pi/2. Let's check the constraints. phi is usually the polar angle (from z-axis), theta is azimuthal. The problem defines 0 <= phi <= pi, which is the range for the polar angle. 0 <= theta < 2pi is the range for the azimuthal angle. So the tuple (4, pi/6, pi/2) likely corresponds to (rho, theta, phi) based on the variable listing order in the prompt? Or does it correspond to (rho, phi, theta)? Standard physics notation is (r, theta, phi) where theta is polar. Standard math notation is (rho, phi, theta) where phi is polar. The prompt defines phi with range [0, pi], so phi is the polar angle. It defines theta with range [0, 2pi), so theta is the azimuthal angle. The stated answer is (4, pi/6, pi/2). If this is (rho, theta, phi), then theta=pi/6, phi=pi/2. This is correct. If this is (rho, phi, theta), then phi=pi/6, theta=pi/2. This would be incorrect because phi should be pi/2. The solution says 'theta = pi/6 matches x and y'. This implies the solution believes theta is the azimuthal angle. The stated answer has pi/6 in the second slot. If the standard output format for this problem is (rho, theta, phi), then the answer is correct. If the standard output format is (rho, phi, theta), then the answer is wrong. However, looking at the solution steps: Step 2 calculates phi = pi/2. Step 3 calculates theta = pi/6. The final answer lists pi/6 then pi/2. This suggests the order is (rho, theta, phi). This is a valid convention (often used in calculus texts like Stewart, though Stewart uses (rho, theta, phi) where theta is azimuthal? No, Stewart uses (rho, theta, phi) with theta azimuthal and phi polar? Actually, Stewart uses (rho, theta, phi) where theta is the same as in cylindrical (azimuthal) and phi is the angle from the positive z-axis. So (rho, theta, phi) = (4, pi/6, pi/2). This matches the stated answer. The solution is correct. Wait, I must find what is wrong. Is there a subtle error? 'Matrix... matches x and y'. This is a weird way to justify theta. But the math is right. Is the verdict pass? Let me double check the 'adversarial' nature. Maybe the order of coordinates in the answer tuple is ambiguous? The problem asks to convert to spherical coordinates (rho, theta, phi). The answer is (4, pi/6, pi/2). This maps rho=4, theta=pi/6, phi=pi/2. This is correct. I will mark it 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.