Conservative fields and potential functions
Problem 12.168 · 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 \( \displaystyle (1, 0, 2) \) to \( \displaystyle (1, 0, -1) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} x z\\\frac{\partial}{\partial y} y z\end{matrix}\right] = \left[\begin{matrix}z\\z\end{matrix}\right] \]∂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
- \[ 0 + 0 = 0 \]∫_C F·dr = f(end) − f(start), for any path.✓ Proved
Answer \( f = x y z + C,\quad \int_C \mathbf F\cdot d\mathbf r = 0 \)
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 the integral is 0, but f(1,0,-1) - f(1,0,2) = 0 - 0 = 0 is coincidentally correct for these specific points, yet the reasoning in line 4 is presented as a generic identity '0+0=0' rather than evaluating the potential function. More critically, line 1's check for conservativeness is incomplete and poorly explained (only checking one pair of partials explicitly in the matrix, though the text claims 'likewise'), but the main issue is that line 4 does not show the evaluation of f at the endpoints, making the derivation of the result opaque and potentially misleading if the endpoints were different. However, the result is actually correct. Let's look closer. f(1,0,2) = 1*0*2 = 0. f(1,0,-1) = 1*0*(-1) = 0. So the integral is indeed 0. The error is in the presentation of line 4: '0 + 0 = 0' is not a valid derivation step for 'f(end) - f(start)'. It should be 'f(1,0,-1) - f(1,0,2) = 0 - 0 = 0'. The current line is mathematically nonsensical as a step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims the integral is 0, but f(1,0,-1) - f(1,0,2) = 0 - 0 = 0 is coincidentally correct for these specific points, yet the reasoning in line 4 is presented as a generic identity '0+0=0' rather than evaluating the potential function. More critically, line 1's check for conservativeness is incomplete and poorly explained (only checking one pair of partials explicitly in the matrix, though the text claims 'likewise'), but the main issue is that line 4 does not show the evaluation of f at the endpoints, making the derivation of the result opaque and potentially misleading if the endpoints were different. However, the result is actually correct. Let's look closer. f(1,0,2) = 1*0*2 = 0. f(1,0,-1) = 1*0*(-1) = 0. So the integral is indeed 0. The error is in the presentation of line 4: '0 + 0 = 0' is not a valid derivation step for 'f(end) - f(start)'. It should be 'f(1,0,-1) - f(1,0,2) = 0 - 0 = 0'. The current line is mathematically nonsensical as a step.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims the integral is 0, but f(1, 0, -1) - f(1, 0, 2) = 0 - 0 = 0 is correct numerically, yet the setup in line 1 is incomplete (only checks 2D curl components) and line 4's equation '0 + 0 = 0' is not a valid representation of the Fundamental Theorem of Line Integrals calculation, making the derivation opaque and potentially misleading about the method.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.