Home›Calculus 3›The divergence theorem›Problem 12.175

The divergence theorem

Problem 12.175 · easy

Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(y\right)\mathbf j + \left(x y\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 3] \times [0, 1] \times [0, 1] \).

\[ \frac{d}{d x} x^{2} + \frac{d}{d y} y + \frac{\partial}{\partial z} x y = 2 x + 1 \]
div F.✓ Proved
Flux out of a closed surface = ∭ div F dV over the solid inside it.
Reviewed
\[ \int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{3} \left(2 x + 1\right)\, dx\, dy\, dz = 12 \]
The triple integral.✓ Proved

Answer \( 12 \)

✓ 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 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 sets up the triple integral with the correct bounds for the specified box.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and sets up the triple integral with the correct bounds for the specified box.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the divergence theorem, computes the divergence accurately, and evaluates the triple integral over the specified box. The steps are logically sound and the final answer is correct.
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.