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The divergence theorem

Problem 12.292 · medium

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

\[ \frac{d}{d y} \left(- 2 y\right) + \frac{d}{d z} z + \frac{\partial}{\partial x} y z = -1 \]
div F.✓ Proved
Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1} \left(- \rho^{2} \sin{\left(\phi \right)}\right)\, d\rho\, d\phi\, d\theta = - \frac{4 \pi}{3} \]
The triple integral.✓ Proved

Answer \( - \frac{4 \pi}{3} \)

✓ 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 as -1, and evaluates the resulting triple integral over the unit ball to obtain -4pi/3.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the Divergence Theorem, computes the divergence as -1, and evaluates the resulting triple integral over the unit ball to obtain -4pi/3.
gpt-oss:20b: pass 2026-10-11
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11

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.