Flux across a plane curve
Problem 12.424 · medium
Find the flux of \( \displaystyle \mathbf F = \left\langle x, y \right\rangle \) across the segment from (0, 0) to (1, 2).
- With r(t) = <t, 2*t>, the normal n ds = ⟨y′, −x′⟩ dt (to the right of the direction of travel), so flux = ∫ (P y′ − Q x′) dt.Reviewed
- \[ - \frac{d}{d t} t \left. y \right|_{\substack{ x=t\\ y=2 t }} + \frac{d}{d t} 2 t \left. x \right|_{\substack{ x=t\\ y=2 t }} = 0 \]P y′ − Q x′.✓ Proved
- \[ \int\limits_{0}^{1} 0\, dt = 0 \]Integrate.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ 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 | F · n summed over a 20,000-segment polyline |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the parameterization and the formula for flux across a curve in 2D. The calculation is correct, yielding 0.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the parameterization and the formula for flux across a curve in 2D. The calculation is correct, yielding 0.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the flux integral formula for a 2D vector field across a curve. The normal vector choice corresponds to the standard outward/right normal, and the calculation is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the parameterization and the formula for flux across a curve in the plane. The calculation is correct, yielding 0.gpt-oss:20b: pass 2026-10-15
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/flux_plane, checked 2026-10-18 with SymPy 1.14.0.