Home›Calculus 3›Flux across a plane curve›Problem 12.334

Flux across a plane curve

Problem 12.334 · medium

Find the flux of \( \displaystyle \mathbf F = \left\langle x y, y^{2} \right\rangle \) across the parabola y = x² from (0, 0) to (1, 1).

With r(t) = <t, t**2>, the normal n ds = ⟨y′, −x′⟩ dt (to the right of the direction of travel), so flux = ∫ (P y′ − Q x′) dt.
Reviewed
\[ - \frac{d}{d t} t \left. y^{2} \right|_{\substack{ x=t\\ y=t^{2} }} + \frac{d}{d t} t^{2} \left. x y \right|_{\substack{ x=t\\ y=t^{2} }} = t^{4} \]
P y′ − Q x′.✓ Proved
\[ \int\limits_{0}^{1} t^{4}\, dt = \frac{1}{5} \]
Integrate.✓ Proved

Answer \( \frac{1}{5} \)

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The normal vector formula used, <y', -x'>, points to the right of the direction of travel. For the curve y=x^2 from (0,0) to (1,1), 'right' corresponds to the interior of the parabola (down/left relative to the curve's concavity), which is typically not the standard orientation for 'flux across a curve' unless specified. More critically, the standard flux integral formula for a vector field <P, Q> across a curve parameterized by r(t) = <x(t), y(t)> with the normal pointing to the right is indeed \int (P y' - Q x') dt. However, the problem does not specify the orientation of the normal (inward vs outward, or left vs right). In standard calculus contexts, 'flux across a curve' usually implies the normal pointing to the left (standard outward for closed curves, or consistent with the Frenet frame). If the standard convention (normal to the left, <-y', x'>) is assumed, the sign would be flipped. But the bigger issue is ambiguity: without specifying 'right' or 'left', the answer is ambiguous. However, looking at the provided solution, it explicitly chooses 'right'. Is this wrong? Usually, flux is defined with the normal pointing to the left of the tangent vector for consistency with Green's Theorem orientation (counter-clockwise). Let's check the standard definition. Flux = \int F . n ds. If n is the unit normal to the left, n = <-y', x'> / |r'|. Then F . n ds = (P(-y') + Q(x')) dt. The solution uses <y', -x'>, which is the normal to the right. This yields -(P(-y') + Q(x')) = P y' - Q x'. The solution calculates this. Is the 'right' normal the standard? No, the 'left' normal is standard. Therefore, the sign is likely wrong based on standard conventions, making the result -1/5 instead of 1/5, or the problem implies a specific orientation not stated. Given the ambiguity, stating 'to the right' as a fact without justification is misleading/error-prone. However, often 'flux across' implies the normal pointing away from the origin or similar. Let's look at the vector field. F = <xy, y^2>. At (0.5, 0.25), F = <0.125, 0.0625>. Tangent is <1, 1>. Left normal is <-1, 1>. Dot product is -0.125 + 0.0625 < 0. Right normal is <1, -1>. Dot product is 0.125 - 0.0625 > 0. The solution gets positive 1/5. If the standard is left-normal, the answer should be negative. The solution arbitrarily picks 'right'. This is an error in convention application.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14
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.