The divergence theorem
Problem 12.266 · easy
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x\right)\mathbf i + \left(y^{3}\right)\mathbf j + \left(0\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
- \[ \frac{d}{d z} 0 + \frac{d}{d x} x + \frac{d}{d y} y^{3} = 3 y^{2} + 1 \]div F.✓ Proved
- Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2} \rho^{2} \left(3 \rho^{2} \sin^{2}{\left(\phi \right)} \sin^{2}{\left(\theta \right)} + 1\right) \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = \frac{544 \pi}{15} \]The triple integral.✓ Proved
Answer \( \frac{544 \pi}{15} \)
Lines: 2 proved, 1 reviewed. 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.
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, computes the divergence, sets up the triple integral in spherical coordinates with correct bounds and Jacobian, and obtains the correct result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the divergence theorem, computes the divergence, sets up the triple integral in spherical coordinates with correct bounds and Jacobian, and obtains the correct result.gpt-oss:20b: fail (error) 2026-10-09 — The integrand in spherical coordinates is incorrect: the term 3y² becomes 3ρ⁴sin³φ sin²θ, not 3ρ²sin²φ sin²θ. This mistake invalidates the computed flux.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the Divergence Theorem. The divergence calculation is correct, and the conversion to spherical coordinates (including the Jacobian and bounds) is accurate, leading to the correct result.
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 2026-10-09 with SymPy 1.14.0.