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The divergence theorem

Problem 12.267 · medium

Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(- 2 y\right)\mathbf j + \left(0\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 1 \).

\[ \frac{d}{d z} 0 + \frac{d}{d x} x^{2} + \frac{d}{d y} \left(- 2 y\right) = 2 x - 2 \]
div F.✓ Proved
Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1} \rho^{2} \left(2 \rho \sin{\left(\phi \right)} \cos{\left(\theta \right)} - 2\right) \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = - \frac{8 \pi}{3} \]
The triple integral.✓ Proved

Answer \( - \frac{8 \pi}{3} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and sets up the triple integral in spherical coordinates with correct bounds and integrand.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and sets up the triple integral in spherical coordinates with correct bounds and integrand.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09

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.