Conservative fields and potential functions
Problem 12.166 · easy
Is \( \displaystyle \mathbf F = \left(y e^{x y}\right)\mathbf i + \left(x e^{x y} + x\right)\mathbf j \) conservative? If so, find a potential function \( \displaystyle f \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x e^{x y} + x\right)\\\frac{\partial}{\partial y} y e^{x y}\end{matrix}\right] = \left[\begin{matrix}x y e^{x y} + e^{x y} + 1\\x y e^{x y} + e^{x y}\end{matrix}\right] \]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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly computes the partial derivatives for the curl test. It calculates ∂Q/∂x as ∂P/∂x and ∂P/∂y as ∂Q/∂y, rather than comparing ∂Q/∂x with ∂P/∂y. The correct calculation shows ∂Q/∂x = ∂P/∂y, so the field is conservative.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly computes the partial derivatives for the curl test. It calculates ∂Q/∂x as ∂P/∂x and ∂P/∂y as ∂Q/∂y, rather than comparing ∂Q/∂x with ∂P/∂y. The correct calculation shows ∂Q/∂x = ∂P/∂y, so the field is conservative.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly computes the partial derivatives for the curl test. It calculates ∂Q/∂x as ∂/∂x(xe^{xy}+x) and ∂P/∂y as ∂/∂y(ye^{xy}), but the standard test requires comparing ∂Q/∂x with ∂P/∂y where F = P i + Q j. Here P = ye^{xy} and Q = xe^{xy}+x. The correct comparison is ∂Q/∂x = ye^{xy} + xye^{xy} + 1 and ∂P/∂y = e^{xy} + xye^{xy}. These are equal, so the field IS conservative. The solution swapped the roles or computed the wrong derivatives entirely, leading to the wrong conclusion.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.