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The divergence theorem

Problem 12.234 · easy

Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x y\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(x y\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).

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

Answer \( 0 \)

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: pass
qwen3.6:27b-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 of 0 due to symmetry.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — 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 of 0 due to symmetry.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The divergence calculation is incorrect; div F = y + 2y + 0 = 3y, not 3y derived from the wrong partials (specifically, the z-component derivative is 0, not y). Furthermore, the integrand in spherical coordinates is wrong: 3y = 3*rho*sin(phi)*sin(theta), but the solution uses 3*rho^3*sin(phi)^2*sin(theta). Although the final answer 0 is correct by symmetry, the intermediate steps contain significant algebraic and conversion errors.

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 2026-10-07 with SymPy 1.14.0.