The divergence theorem
Problem 12.201 · easy
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(x^{2}\right)\mathbf i + \left(y^{2}\right)\mathbf j + \left(x y\right)\mathbf k \) across the surface of the box \( \displaystyle [0, 1] \times [0, 3] \times [0, 2] \).
- \[ \frac{d}{d x} x^{2} + \frac{d}{d y} y^{2} + \frac{\partial}{\partial z} x y = 2 x + 2 y \]div F.✓ Proved
- Flux out of a closed surface = ∭ div F dV over the solid inside it.
- \[ \int\limits_{0}^{2}\int\limits_{0}^{3}\int\limits_{0}^{1} \left(2 x + 2 y\right)\, dx\, dy\, dz = 24 \]The triple integral.✓ Proved
Answer \( 24 \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The divergence is incorrectly computed: ∂(xy)/∂z = 0, not 2y. The rest of the reasoning is correct once the divergence is fixed.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 given box. The final result is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The divergence is incorrectly computed: ∂(xy)/∂z = 0, not 2y. The rest of the reasoning is correct once the divergence is fixed.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and sets up the triple integral with the correct bounds for the given box. The final result is correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the Divergence Theorem, computes the divergence accurately, and sets up the triple integral with the correct bounds for the given box. The final result is correct.
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.