Home›Calculus 3›Flux across a plane curve›Problem 12.426

Flux across a plane curve

Problem 12.426 · medium

Find the flux of \( \displaystyle \mathbf F = \left\langle x y, y^{2} \right\rangle \) 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
\[ - \frac{d}{d t} t \left. y^{2} \right|_{\substack{ x=t\\ y=2 t }} + \frac{d}{d t} 2 t \left. x y \right|_{\substack{ x=t\\ y=2 t }} = 0 \]
P y′ − Q x′.✓ Proved
\[ \int\limits_{0}^{1} 0\, dt = 0 \]
Integrate.✓ Proved

Answer \( 0 \)

Lines: 2 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the parametrization and the formula for flux across a curve in 2D. The calculation is correct, and the result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the parametrization and the formula for flux across a curve in 2D. The calculation is correct, and the result matches the stated answer.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly parameterizes the segment, identifies the correct integrand for flux (P dy - Q dx), and computes the integral accurately.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The normal vector formula used, <y', -x'>, corresponds to the left normal, not the right normal as claimed. For a curve moving from (0,0) to (1,2), the right normal points into the fourth quadrant (positive x, negative y), whereas <2, -1> points into the first quadrant. While the final numerical answer happens to be correct because the integral of the resulting expression is zero, the stated geometric interpretation of the normal vector is incorrect.
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.