Home›Calculus 3›Flux across a plane curve›Problem 12.396

Flux across a plane curve

Problem 12.396 · medium

Find the flux of \( \displaystyle \mathbf F = \left\langle x y, y^{2} \right\rangle \) across the segment from (0, 0) to (1, 3).

With r(t) = <t, 3*t>, the normal n ds = ⟨y′, −x′⟩ dt (to the right of the direction of travel), so flux = ∫ (P y′ − Q x′) dt.
Reviewed
\[ - \frac{d}{d t} t \left. y^{2} \right|_{\substack{ x=t\\ y=3 t }} + \frac{d}{d t} 3 t \left. x y \right|_{\substack{ x=t\\ y=3 t }} = 0 \]
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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the normal vector for flux across a 2D curve and sets up the integral properly. The calculation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the normal vector for flux across a 2D curve and sets up the integral properly. The calculation is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameterization and the formula for flux across a curve in 2D. The calculation is correct.
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 2026-10-18 with SymPy 1.14.0.