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Divergence and curl

Problem 12.127 · easy

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

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

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

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 should result in a scalar and a 3D vector, respectively. The solution computes the correct components but fails to assemble them into the correct final format or explicitly state the divergence as a scalar separate from the curl vector.

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 should result in a scalar and a 3D vector, respectively. The solution computes the correct components but fails to assemble them into the correct final format or explicitly state the divergence as a scalar separate from the curl vector.
gpt-oss:20b: pass 2026-09-27
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 computes the correct components but fails to present them in the correct format or combine them into a single coherent answer as implied by the problem statement.
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.