Home›Calculus 3›Divergence and curl›Problem 12.78

Divergence and curl

Problem 12.78 · easy

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

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

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

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. How we verify

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 computes them correctly but fails to combine them into the requested format or explain the discrepancy. Additionally, the first line computes the divergence but labels it with 'z' which is confusing, and the final answer structure is completely wrong for the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution computes them correctly but fails to combine them into the requested format or explain the discrepancy. Additionally, the first line computes the divergence but labels it with 'z' which is confusing, and the final answer structure is completely wrong for the problem statement.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution calculates them correctly but fails to combine them into the correct final format or explain the discrepancy. Additionally, the first line labels the divergence calculation as just 'z', omitting the scalar result itself in the text description, though the equation is marked proved.
gpt-oss:20b: pass 2026-09-27
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 fails to combine the results into a coherent final answer matching the problem statement.
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 2026-09-28 with SymPy 1.14.0.