Home›Calculus 3›The divergence theorem›Problem 12.200

The divergence theorem

Problem 12.200 · easy

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

\[ \frac{d}{d y} y^{2} + \frac{d}{d z} z + \frac{\partial}{\partial x} x y = 3 y + 1 \]
div F.✓ Proved
Flux out of a closed surface = ∭ div F dV over the solid inside it.
Reviewed
\[ \int\limits_{0}^{3}\int\limits_{0}^{3}\int\limits_{0}^{3} \left(3 y + 1\right)\, dx\, dy\, dz = \frac{297}{2} \]
The triple integral.✓ Proved

Answer \( \frac{297}{2} \)

✓ 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 evaluates the triple integral over the specified box.

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.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the divergence theorem. The divergence is calculated correctly, and the triple integral over the specified box yields the correct result.

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.