Flux across a plane curve
Problem 12.339 · medium
Find the flux of \( \displaystyle \mathbf F = \left\langle x^{2}, y \right\rangle \) across the segment from (0, 0) to (1, 1).
- With r(t) = <t, t>, the normal n ds = ⟨y′, −x′⟩ dt (to the right of the direction of travel), so flux = ∫ (P y′ − Q x′) dt.
- \[ \frac{d}{d t} t \left. x^{2} \right|_{\substack{ x=t\\ y=t }} - \frac{d}{d t} t \left. y \right|_{\substack{ x=t\\ y=t }} = t \left(t - 1\right) \]P y′ − Q x′.✓ Proved
- \[ \int\limits_{0}^{1} t \left(t - 1\right)\, dt = - \frac{1}{6} \]Integrate.✓ Proved
Answer \( - \frac{1}{6} \)
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: passqwen3.6:27b-mlx: fail (error) — The normal vector formula used, <y', -x'>, points to the left of the direction of travel, not the right. For a curve moving in the first quadrant, the right normal should have positive x and negative y components (e.g., <1, -1>), whereas <y', -x'> yields <-1, 1>. This sign error leads to the incorrect negative flux.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The normal vector formula used, <y', -x'>, points to the left of the direction of travel, not the right. For a curve moving in the first quadrant, the right normal should have positive x and negative y components (e.g., <1, -1>), whereas <y', -x'> yields <-1, 1>. This sign error leads to the incorrect negative flux.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vector used, <y', -x'>, points to the left of the direction of travel, not the right. For a curve moving in the first quadrant, <1, -1> points into the fourth quadrant (left), whereas the right normal is <-1, 1>. This sign error leads to the incorrect negative flux.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The normal vector formula used, <y', -x'>, points to the right of the direction of travel. For a path from (0,0) to (1,1), 'right' corresponds to the vector <1, -1>, which points into the fourth quadrant. Standard flux problems usually imply the normal pointing to the left (or outward for closed curves), or require explicit specification. More critically, the standard definition of flux across a curve C is ∫ F · n ds. If n is the unit normal, ds = |r'| dt. The vector <y', -x'> has magnitude |r'|, so <y', -x'> dt is indeed a normal vector scaled by arc length element, but it is the 'right-hand' normal. The 'left-hand' normal (standard for counter-clockwise or standard orientation) is <-y', x'>. Using the right-hand normal gives the negative of the standard flux. Without explicit instruction to use the right-hand normal, this is likely an error in sign convention, leading to the negative answer. However, checking the math: F = <x^2, y>. r(t) = <t, t>. r'(t) = <1, 1>. Left normal n = <-1, 1>. F · n = <t^2, t> · <-1, 1> = -t^2 + t. Integral from 0 to 1 of (t - t^2) dt = [t^2/2 - t^3/3]_0^1 = 1/2 - 1/3 = 1/6. The solution uses the right normal <1, -1>, getting t^2 - t, integral -1/6. Unless the problem specifies 'flux to the right', the standard convention is the left normal (or outward for closed loops). Given the ambiguity, stating the normal is 'to the right' is a choice, but usually 'flux across' implies the standard orientation (left). If the answer key is -1/6, the problem likely implies the right normal, but the solution text asserts 'n ds = <y', -x'> dt (to the right...)' as a definition. This is a valid definition for *a* normal, but is it the *correct* one for the problem? Usually, flux is defined with the normal obtained by rotating the tangent 90 degrees counter-clockwise (left). Rotating <1,1> 90 deg CCW gives <-1, 1>. The solution uses CW rotation. This is a sign error relative to standard convention.gpt-oss:20b: pass 2026-10-14
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.