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Conservative fields and potential functions

Problem 12.227 · easy

Is \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(x z + x\right)\mathbf j + \left(x y\right)\mathbf k \) conservative? If so, find a potential function \( \displaystyle f \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x z + x\right)\\\frac{\partial}{\partial y} y z\end{matrix}\right] = \left[\begin{matrix}z + 1\\z\end{matrix}\right] \]
Compare ∂Q/∂x with ∂P/∂y.✓ Proved
They differ, so curl F ≠ 0 and F is not conservative: no potential exists.

Answer \( \text{not conservative} \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	∂Q/∂x − ∂P/∂y recomputed: not identically zero

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly checks 2D curl conditions (∂Q/∂x vs ∂P/∂y) for a 3D vector field, ignoring the z-component. The field is actually conservative (∇×F = 0), so the conclusion is wrong.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly checks 2D curl conditions (∂Q/∂x vs ∂P/∂y) for a 3D vector field, ignoring the z-component. The field is actually conservative (∇×F = 0), so the conclusion is wrong.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly compares ∂Q/∂x and ∂P/∂y to determine conservativeness, ignoring the other necessary conditions (∂R/∂y = ∂Q/∂z and ∂P/∂z = ∂R/∂x). In fact, the field is conservative, and the potential function is f = xyz + x.
gpt-oss:20b: pass 2026-10-07

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/conservative_fields, checked 2026-10-07 with SymPy 1.14.0.