Divergence and curl
Problem 12.139 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 z^{2}, 2 y, x \rangle \).
- \[ \frac{d}{d z} x + \frac{d}{d y} 2 y + \frac{d}{d x} 2 z^{2} = 2 \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} x - \frac{d}{d z} 2 y\\- \frac{d}{d x} x + \frac{d}{d z} 2 z^{2}\\\frac{d}{d x} 2 y - \frac{d}{d y} 2 z^{2}\end{matrix}\right] = \left[\begin{matrix}0\\4 z - 1\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = 2,\quad \nabla\times\mathbf{F} = \langle 0, 4 z - 1, 0 \rangle \)
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | difference quotients of each component at (0.3, 0.7, 1.1) agree |
Reviewers
gpt-oss:20b: fail (misleading) — The divergence computation uses incorrect partial derivatives (∂/∂x of the first component and ∂/∂z of the third), yet the numerical result happens to be correct. This misleads a student about the proper formula for divergence.qwen3.6:27b-mlx: fail (error) — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3D curl components, but the final answer format is inconsistent with standard vector calculus definitions and the problem statement.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3D curl components, but the final answer format is inconsistent with standard vector calculus definitions and the problem statement.gpt-oss:20b: fail (misleading) 2026-09-29 — The divergence computation uses incorrect partial derivatives (∂/∂x of the first component and ∂/∂z of the third), yet the numerical result happens to be correct. This misleads a student about the proper formula for divergence.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution computes them separately but fails to combine them into the correct final format or explain the discrepancy.gpt-oss:20b: pass 2026-09-29
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/divergence_curl, checked 2026-09-29 with SymPy 1.14.0.