The divergence theorem
Problem 12.294 · easy
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x y\right)\mathbf i + \left(y^{3}\right)\mathbf j + \left(z^{3}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).
- \[ \frac{d}{d y} y^{3} + \frac{d}{d z} z^{3} + \frac{\partial}{\partial x} x y = 3 y^{2} + y + 3 z^{2} \]div F.✓ Proved
- Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1} \rho^{2} \left(3 \rho^{2} \sin^{2}{\left(\phi \right)} \sin^{2}{\left(\theta \right)} + 3 \rho^{2} \cos^{2}{\left(\phi \right)} + \rho \sin{\left(\phi \right)} \sin{\left(\theta \right)}\right) \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = \frac{8 \pi}{5} \]The triple integral.✓ Proved
Answer \( \frac{8 \pi}{5} \)
✓ 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | the surface integral F·n dS over the sphere, computed directly |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the divergence theorem. The divergence calculation is correct, and the conversion to spherical coordinates and subsequent integration yield the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the divergence theorem. The divergence calculation is correct, and the conversion to spherical coordinates and subsequent integration yield the correct result.gpt-oss:20b: pass 2026-10-11gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the Divergence Theorem. The divergence calculation is correct, and the triple integral in spherical coordinates is set up and evaluated correctly to yield 8π/5.
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/divergence_theorem, checked 2026-10-11 with SymPy 1.14.0.