Flux across a plane curve
Problem 12.333 · medium
Find the flux of \( \displaystyle \mathbf F = \left\langle x + y, x - y \right\rangle \) across the circle of radius 1, counterclockwise (outward normal).
- With r(t) = <cos(t), sin(t)>, the normal n ds = ⟨y′, −x′⟩ dt (to the right of the direction of travel), so flux = ∫ (P y′ − Q x′) dt.
- \[ \frac{d}{d t} \sin{\left(t \right)} \left. x + y \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)} }} - \frac{d}{d t} \cos{\left(t \right)} \left. x - y \right|_{\substack{ x=\cos{\left(t \right)}\\ y=\sin{\left(t \right)} }} = \sqrt{2} \sin{\left(2 t + \frac{\pi}{4} \right)} \]P y′ − Q x′.✓ Proved
- \[ \int\limits_{0}^{2 \pi} \sqrt{2} \sin{\left(2 t + \frac{\pi}{4} \right)}\, dt = 0 \]Integrate.✓ Proved
Answer \( 0 \)
Lines: 2 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The normal vector used is incorrect: for a unit circle traversed counter‑clockwise, the outward normal is <cos t, sin t>, not <y′, −x′> = <−sin t, −cos t>. Consequently the flux formula P y′−Q x′ is for the inward normal, so the setup is wrong even though the integral happens to vanish.qwen3.6:27b-mlx: fail (error) — The normal vector formula is incorrect for the outward normal. For a counterclockwise traversal, the outward normal is <x', y'>, not <y', -x'> (which is the inward normal). This sign error leads to the wrong integrand, although the final answer happens to be 0 due to symmetry.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The normal vector formula is incorrect for the outward normal. For a counterclockwise traversal, the outward normal is <x', y'>, not <y', -x'> (which is the inward normal). This sign error leads to the wrong integrand, although the final answer happens to be 0 due to symmetry.gpt-oss:20b: fail (error) 2026-10-17 — The normal vector used is incorrect: for a unit circle traversed counter‑clockwise, the outward normal is <cos t, sin t>, not <y′, −x′> = <−sin t, −cos t>. Consequently the flux formula P y′−Q x′ is for the inward normal, so the setup is wrong even though the integral happens to vanish.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vector formula used is incorrect for the outward normal. For a counterclockwise parametrization, the outward normal is <x', y'>, not <y', -x'> (which is the inward normal). This sign error leads to the wrong integrand, although the final answer happens to be 0 due to symmetry.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The normal vector formula is incorrect for the outward normal; for a counterclockwise traversal, the outward normal is <x', y'>, not <y', -x'> (which points inward). Consequently, the integrand should be P x' + Q y', not P y' - Q x'. Although the final answer happens to be correct due to symmetry, the method described is wrong.gpt-oss:20b: fail (misleading) 2026-10-14 — The solution claims to use the outward normal but the vector <y′,−x′> is the inward normal for a counter‑clockwise traversal. Although the integral still evaluates to zero, the reasoning misidentifies the normal direction, which could mislead a student about how to set up flux integrals for closed curves.
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 2026-10-17 with SymPy 1.14.0.