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Divergence and curl

Problem 12.137 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - x y z, - 3 z, 2 x y \rangle \).

\[ \frac{d}{d y} \left(- 3 z\right) + \frac{\partial}{\partial z} 2 x y + \frac{\partial}{\partial x} \left(- x y z\right) = - y z \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- 3 z\right) + \frac{\partial}{\partial y} 2 x y\\- \frac{\partial}{\partial x} 2 x y + \frac{\partial}{\partial z} \left(- x y z\right)\\\frac{d}{d x} \left(- 3 z\right) - \frac{\partial}{\partial y} \left(- x y z\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3\\- x y - 2 y\\x z\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = - y z,\quad \nabla\times\mathbf{F} = \langle 2 x + 3, - x y - 2 y, x z \rangle \)

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: pass
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 3 components of the curl, but the final answer format is incorrect (likely including the divergence as a 4th component or a formatting error).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3 components of the curl, but the final answer format is incorrect (likely including the divergence as a 4th component or a formatting error).
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3 components of the curl, but the final answer format is incorrect (likely an artifact of the CAS output format including an extra row or misinterpreting the problem as 4D). Additionally, the divergence is a scalar, not part of the curl matrix, yet the stated answer mixes them or presents them in a non-standard structure.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.