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Divergence and curl

Problem 12.123 · easy

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

\[ \frac{d}{d x} \left(- 3 x\right) + \frac{d}{d y} \left(- y^{2}\right) + \frac{\partial}{\partial z} 3 x z = 3 x - 2 y - 3 \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- y^{2}\right) + \frac{\partial}{\partial y} 3 x z\\\frac{d}{d z} \left(- 3 x\right) - \frac{\partial}{\partial x} 3 x z\\- \frac{d}{d y} \left(- 3 x\right) + \frac{d}{d x} \left(- y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}0\\- 3 z\\0\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = 3 x - 2 y - 3,\quad \nabla\times\mathbf{F} = \langle 0, - 3 z, 0 \rangle \)

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: pass
qwen3.6:27b-mlx: fail (error) — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field are a scalar and a 3D vector, respectively. The solution calculates the correct components but fails to assemble them into the correct format or explain the discrepancy with the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field are a scalar and a 3D vector, respectively. The solution calculates the correct components but fails to assemble them into the correct format or explain the discrepancy with the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix containing both the divergence and the curl, but the solution only computes the divergence and curl separately. The solution fails to combine them into the final stated format, and the stated answer itself is structurally ambiguous (mixing a scalar divergence with a vector curl in a single column vector without clear labeling or context). Furthermore, the problem asks for 'divergence and curl', which are typically returned as a scalar and a vector respectively, not concatenated into a single 4-element vector unless specified. The provided 'Stated answer' is likely incorrect in format or interpretation.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.