Conservative fields and potential functions
Problem 12.255 · medium
Show that \( \displaystyle \mathbf F = \left(y z e^{x}\right)\mathbf i + \left(z e^{x}\right)\mathbf j + \left(y e^{x}\right)\mathbf k \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (2, 1, 0) \) to \( \displaystyle (1, 1, 0) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} z e^{x}\\\frac{\partial}{\partial y} y z e^{x}\end{matrix}\right] = \left[\begin{matrix}z e^{x}\\z e^{x}\end{matrix}\right] \]∂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.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} y z e^{x}\\\frac{\partial}{\partial y} y z e^{x}\\\frac{\partial}{\partial z} y z e^{x}\end{matrix}\right] = \left[\begin{matrix}y z e^{x}\\z e^{x}\\y e^{x}\end{matrix}\right] \]f = y*z*exp(x) has gradient F.✓ Proved
- \[ 0 + 0 = 0 \]∫_C F·dr = f(end) − f(start), for any path.✓ Proved
Answer \( f = y z e^{x} + C,\quad \int_C \mathbf F\cdot d\mathbf r = 0 \)
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(1,1,0) - f(2,1,0) = 0 - 0 = 0 is correct only if the z-coordinate is 0 in both points, which it is. Wait, let's re-evaluate. f(x,y,z) = yze^x. Start: (2,1,0) -> f=1*0*e^2=0. End: (1,1,0) -> f=1*0*e^1=0. So the answer 0 is correct. However, the sentence in step 1 says 'F is conservative' based on checking partials. It checks P_y = Q_x? P=yz e^x, Q=z e^x. P_y = z e^x, Q_x = z e^x. Correct. But it doesn't explicitly check all three curl components, though it says 'and likewise'. The main issue is Step 4: '0 + 0 = 0' is a nonsensical equation for the Fundamental Theorem of Line Integrals. It should be f(end) - f(start). While the numerical result is coincidentally 0, the representation '0+0=0' is misleading and doesn't show the application of the theorem properly. Actually, looking closer, the prompt asks to review the *sentences*. Step 4 is an equation marked proved, but the comment says 'integral = f(end) - f(start)'. The equation itself '0+0=0' is algebraically true but semantically empty/wrong representation of the theorem application. Is it an error? It's misleading. A student would learn that the theorem is '0+0=0'. That is bad. Let's call it misleading.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution claims the integral is 0, but f(1,1,0) - f(2,1,0) = 0 - 0 = 0 is correct only if the z-coordinate is 0 in both points, which it is. Wait, let's re-evaluate. f(x,y,z) = yze^x. Start: (2,1,0) -> f=1*0*e^2=0. End: (1,1,0) -> f=1*0*e^1=0. So the answer 0 is correct. However, the sentence in step 1 says 'F is conservative' based on checking partials. It checks P_y = Q_x? P=yz e^x, Q=z e^x. P_y = z e^x, Q_x = z e^x. Correct. But it doesn't explicitly check all three curl components, though it says 'and likewise'. The main issue is Step 4: '0 + 0 = 0' is a nonsensical equation for the Fundamental Theorem of Line Integrals. It should be f(end) - f(start). While the numerical result is coincidentally 0, the representation '0+0=0' is misleading and doesn't show the application of the theorem properly. Actually, looking closer, the prompt asks to review the *sentences*. Step 4 is an equation marked proved, but the comment says 'integral = f(end) - f(start)'. The equation itself '0+0=0' is algebraically true but semantically empty/wrong representation of the theorem application. Is it an error? It's misleading. A student would learn that the theorem is '0+0=0'. That is bad. Let's call it misleading.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly calculate the potential function values at the endpoints (2, 1, 0) and (1, 1, 0) to demonstrate that the difference is zero. It merely states the Fundamental Theorem of Line Integrals and asserts the result is 0 without showing the substitution, which is the core of the evaluation task.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.