Conservative fields and potential functions
Problem 12.258 · medium
Show that \( \displaystyle \mathbf F = \left(3 x^{2} + y^{2}\right)\mathbf i + \left(2 x y\right)\mathbf j \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (-1, 2) \) to \( \displaystyle (0, 0) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} 2 x y\\\frac{\partial}{\partial y} \left(3 x^{2} + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 y\\2 y\end{matrix}\right] \]∂Q/∂x = ∂P/∂y: 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.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{3} + x y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{3} + x y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}3 x^{2} + y^{2}\\2 x y\end{matrix}\right] \]f = x**3 + x*y**2 has gradient F.✓ Proved
- \[ 0 + 5 = 5 \]∫_C F·dr = f(end) − f(start), for any path.✓ Proved
Answer \( f = x^{3} + x y^{2} + C,\quad \int_C \mathbf F\cdot d\mathbf r = 5 \)
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 5, but f(0,0) - f(-1,2) = 0 - (-1 + 4) = -3. The arithmetic in step 4 is incorrect.
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 5, but f(0,0) - f(-1,2) = 0 - (-1 + 4) = -3. The arithmetic in step 4 is incorrect.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly verifies the conservative nature of the field, identifies the potential function, and applies the Fundamental Theorem of Line Integrals with correct arithmetic.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.