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Divergence and curl

Problem 12.102 · easy

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

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

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

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 divergence is a scalar and the curl is a 3-vector. The solution correctly computes the components but fails to present them in the standard format (scalar divergence, vector curl) or explain the mismatch with the provided answer format.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution correctly computes the components but fails to present them in the standard format (scalar divergence, vector curl) or explain the mismatch with the provided answer format.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — 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 correctly computes the components but fails to format them into the expected output structure, and the stated answer includes an extra row or incorrect dimensions.
gpt-oss:20b: pass 2026-09-26

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 2026-09-26 with SymPy 1.14.0.