Stokes' theorem
Problem 12.381 · hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle z, x, y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
- Take S to be the flat disk the circle bounds, with upward normal k. Stokes: ∮_C F · dr = ∬_S (curl F) · k dA.
- \[ \left[\begin{matrix}- \frac{d}{d z} x + \frac{d}{d y} y\\- \frac{d}{d x} y + \frac{d}{d z} z\\\frac{d}{d x} x - \frac{d}{d y} z\end{matrix}\right] = \left[\begin{matrix}1\\1\\1\end{matrix}\right] \]curl F.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left. 1 \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = 9 \pi \]The k-component of the curl over the disk, in polar coordinates.✓ Proved
- \[ \int\limits_{0}^{2 \pi} 9 \cos^{2}{\left(t \right)}\, dt = 9 \pi \]The circulation computed directly around C agrees.✓ Proved
Answer \( 9 \pi \)
Lines: 3 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 |
| 4 | ✓ 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 · Δr summed around a 20,000-sided polygon on the circle |
Reviewers
gpt-oss:20b: fail (error) — The curl of F = ⟨z,x,y⟩ is zero, not ⟨1,1,1⟩. The subsequent surface integral and circulation are therefore incorrect.qwen3.6:27b-mlx: pass — The solution correctly applies Stokes' theorem, identifies the curl and normal vector, and computes the surface integral accurately. The final line, while redundant for the requested method, serves as a valid consistency check.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies Stokes' theorem, identifies the curl and normal vector, and computes the surface integral accurately. The final line, while redundant for the requested method, serves as a valid consistency check.gpt-oss:20b: fail (error) 2026-10-17 — The curl of F = ⟨z,x,y⟩ is zero, not ⟨1,1,1⟩. The subsequent surface integral and circulation are therefore incorrect.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies Stokes' theorem, computes the curl, and evaluates the surface integral. The final line confirms the result via direct line integration, which is consistent and correct.gpt-oss:20b: fail (error) 2026-10-15 — The curl of F is incorrectly computed; it should be the zero vector, not <1,1,1>. Consequently the surface integral and the final value of 9π are wrong.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies Stokes' Theorem, computes the curl, and evaluates the surface integral. The final line provides a correct direct verification, though it is not required by the prompt.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/stokes_theorem, checked 2026-10-17 with SymPy 1.14.0.