Conservative fields and potential functions
Problem 12.284 · 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 \( \displaystyle (2, 1, 2) \) to \( \displaystyle (0, -1, 0) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x + z\right)\\\frac{\partial}{\partial y} \left(y + z\right)\end{matrix}\right] = \left[\begin{matrix}1\\1\end{matrix}\right] \]∂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.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y + x z + y z\right)\\\frac{\partial}{\partial y} \left(x y + x z + y z\right)\\\frac{\partial}{\partial z} \left(x y + x z + y z\right)\end{matrix}\right] = \left[\begin{matrix}y + z\\x + z\\x + y\end{matrix}\right] \]f = x*y + x*z + y*z has gradient F.✓ Proved
- \[ -8 + 0 = -8 \]∫_C F·dr = f(end) − f(start), for any path.✓ Proved
Answer \( f = x y + x z + y z + C,\quad \int_C \mathbf F\cdot d\mathbf r = -8 \)
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 F is conservative based on checking only two partial derivative equalities (∂Q/∂x = ∂P/∂y and implicitly ∂R/∂y = ∂Q/∂z? No, the matrix only shows 2x2). For a 3D field, all three curl components must vanish. Furthermore, the text says 'and likewise for the other pairs' but does not show the check for ∂R/∂x vs ∂P/∂z. While the potential function found is correct, the justification for conservativeness is incomplete and technically insufficient as written.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims F is conservative based on checking only two partial derivative equalities (∂Q/∂x = ∂P/∂y and implicitly ∂R/∂y = ∂Q/∂z? No, the matrix only shows 2x2). For a 3D field, all three curl components must vanish. Furthermore, the text says 'and likewise for the other pairs' but does not show the check for ∂R/∂x vs ∂P/∂z. While the potential function found is correct, the justification for conservativeness is incomplete and technically insufficient as written.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims F is conservative based on a 2x2 matrix check, ignoring the third component and the full curl calculation required for 3D vector fields. While the result happens to be correct, the reasoning is incomplete and mathematically insufficient.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.