Stokes' theorem
Problem 12.380 · hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y^{3}, x^{3}, z \right\rangle \), 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.Reviewed
- \[ \left[\begin{matrix}- \frac{d}{d z} x^{3} + \frac{d}{d y} z\\\frac{d}{d z} \left(- y^{3}\right) - \frac{d}{d x} z\\\frac{d}{d x} x^{3} - \frac{d}{d y} \left(- y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\3 x^{2} + 3 y^{2}\end{matrix}\right] \]curl F.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left. 3 x^{2} + 3 y^{2} \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = \frac{243 \pi}{2} \]The k-component of the curl over the disk, in polar coordinates.✓ Proved
- \[ \int\limits_{0}^{2 \pi} \left(81 \sin^{4}{\left(t \right)} + 81 \cos^{4}{\left(t \right)}\right)\, dt = \frac{243 \pi}{2} \]The circulation computed directly around C agrees.✓ Proved
Answer \( \frac{243 \pi}{2} \)
✓ Nihil obstat Lines: 3 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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies Stokes' theorem, computes the curl, and evaluates the surface integral. The direct line integral check is consistent with the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies Stokes' theorem, computes the curl, and evaluates the surface integral. The direct line integral check is consistent with the result.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies Stokes' theorem with the appropriate surface and orientation. The curl calculation and polar integration are correct, and the direct line integral verification confirms the result.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies Stokes' Theorem with the appropriate surface and normal vector. The calculation of the curl and the subsequent double integral in polar coordinates are correct. The direct line integral check is also correct and consistent.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.