Conservative fields and potential functions
Problem 12.160 · medium
Show that \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(x z\right)\mathbf j + \left(x y\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (1, 2, 2) \) to \( \displaystyle (0, 1, 2) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} x z\\\frac{\partial}{\partial y} y z\end{matrix}\right] = \left[\begin{matrix}z\\z\end{matrix}\right] \]∂Q/∂x = ∂P/∂y (and likewise for the other pairs): F is conservative.✓ Proved
- Integrate the first component in x, then fix the 'constant' (a function of the other variables) by matching the other components.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} x y z\\\frac{\partial}{\partial y} x y z\\\frac{\partial}{\partial z} x y z\end{matrix}\right] = \left[\begin{matrix}y z\\x z\\x y\end{matrix}\right] \]f = x*y*z has gradient F.✓ Proved
- \[ -4 + 0 = -4 \]∫_C F·dr = f(end) − f(start), for any path.✓ Proved
Answer \( f = x y z + C,\quad \int_C \mathbf F\cdot d\mathbf r = -4 \)
Lines: 3 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | the line integral computed numerically along two different paths |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims to show F is conservative but only checks 2D curl components (∂Q/∂x = ∂P/∂y), ignoring the 3D conditions involving z. Additionally, the final calculation of the line integral is incorrect: f(0,1,2) - f(1,2,2) = 0 - 4 = -4 is numerically correct, but the intermediate step '-4 + 0' implies f(end)=0 and f(start)=-4, whereas f(start)=4, so it should be 0 - 4.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims to show F is conservative but only checks 2D curl components (∂Q/∂x = ∂P/∂y), ignoring the 3D conditions involving z. Additionally, the final calculation of the line integral is incorrect: f(0,1,2) - f(1,2,2) = 0 - 4 = -4 is numerically correct, but the intermediate step '-4 + 0' implies f(end)=0 and f(start)=-4, whereas f(start)=4, so it should be 0 - 4.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the field as conservative, finds the potential function, and applies the Fundamental Theorem of Line Integrals with the correct evaluation.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.