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The divergence theorem

Problem 12.206 · easy

Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(y\right)\mathbf j + \left(3 z\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 2] \times [0, 3] \times [0, 2] \).

\[ \frac{d}{d x} x^{2} + \frac{d}{d y} y + \frac{d}{d z} 3 z = 2 x + 4 \]
div F.✓ Proved
Flux out of a closed surface = ∭ div F dV over the solid inside it.
Reviewed
\[ \int\limits_{0}^{2}\int\limits_{0}^{3}\int\limits_{0}^{2} \left(2 x + 4\right)\, dx\, dy\, dz = 72 \]
The triple integral.✓ Proved

Answer \( 72 \)

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 flux through the six faces of the box, computed directly

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and evaluates the triple integral over the specified box to obtain the correct flux.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and evaluates the triple integral over the specified box to obtain the correct flux.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The divergence calculation in line 1 is incorrect; the derivative of 3z with respect to z is 3, not 4. This leads to an incorrect final answer.

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 2026-10-06 with SymPy 1.14.0.