The divergence theorem
Problem 12.171 · easy
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(0\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
- \[ \frac{d}{d x} 0 + \frac{d}{d y} y^{2} + \frac{d}{d z} z^{2} = 2 y + 2 z \]div F.✓ Proved
- Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2} \rho^{2} \left(2 \rho \sin{\left(\phi \right)} \sin{\left(\theta \right)} + 2 \rho \cos{\left(\phi \right)}\right) \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = 0 \]The triple integral.✓ Proved
Answer \( 0 \)
✓ 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 is calculated correctly, and the triple integral over the sphere evaluates to 0 due to symmetry (the integrands involving y and z are odd functions integrated over a symmetric domain). The setup and conclusion are sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Divergence Theorem. The divergence is calculated correctly, and the triple integral over the sphere evaluates to 0 due to symmetry (the integrands involving y and z are odd functions integrated over a symmetric domain). The setup and conclusion are sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Divergence Theorem. The divergence is calculated correctly, and the triple integral over the symmetric sphere of the odd functions y and z correctly evaluates to zero.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.