Stokes' theorem
Problem 12.375 · hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle x z, x - y, x y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) 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
- \[ \left[\begin{matrix}\frac{\partial}{\partial y} x y - \frac{\partial}{\partial z} \left(x - y\right)\\- \frac{\partial}{\partial x} x y + \frac{\partial}{\partial z} x z\\- \frac{\partial}{\partial y} x z + \frac{\partial}{\partial x} \left(x - y\right)\end{matrix}\right] = \left[\begin{matrix}x\\x - y\\1\end{matrix}\right] \]curl F.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1} r \left. 1 \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = \pi \]The k-component of the curl over the disk, in polar coordinates.✓ Proved
- \[ \int\limits_{0}^{2 \pi} \left(\frac{\sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)}}{2} + \frac{1}{2}\right)\, dt = \pi \]The circulation computed directly around C agrees.✓ Proved
Answer \( \pi \)
✓ 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 by identifying the surface and normal vector, computing the curl, and evaluating the surface integral. The direct line integral check in step 4 is consistent with the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies Stokes' theorem by identifying the surface and normal vector, computing the curl, and evaluating the surface integral. The direct line integral check in step 4 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 by identifying the surface and normal vector, computing the curl, and evaluating the surface integral. The direct line integral check in step 4 is consistent with the result.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies Stokes' theorem. The curl is calculated correctly, and the surface integral over the unit disk in the z=0 plane yields pi. The direct line integral check is also 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/stokes_theorem, checked 2026-10-17 with SymPy 1.14.0.