The divergence theorem
Problem 12.205 · easy
Use the divergence theorem to find the outward flux of \( \displaystyle \mathbf F = \left(y z\right)\mathbf i + \left(y z\right)\mathbf j + \left(z^{2}\right)\mathbf k \) across the sphere \( \displaystyle x^2 + y^2 + z^2 = 4 \).
- \[ \frac{d}{d z} z^{2} + \frac{\partial}{\partial x} y z + \frac{\partial}{\partial y} y z = 3 z \]div F.✓ Proved
- Flux out of a closed surface = ∭ div F dV over the solid inside it (in spherical coordinates).
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2} 3 \rho^{3} \sin{\left(\phi \right)} \cos{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = 0 \]The triple integral.✓ Proved
Answer \( 0 \)
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 surface integral F·n dS over the sphere, computed directly |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The divergence calculation in step 1 is incorrect; div F is z + 1, not 3z. Consequently, the integral in step 3 is set up with the wrong integrand, leading to an incorrect result of 0 instead of 8π.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The divergence calculation in step 1 is incorrect; div F is z + 1, not 3z. Consequently, the integral in step 3 is set up with the wrong integrand, leading to an incorrect result of 0 instead of 8π.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The divergence calculation in step 1 is incorrect; div F is 2z, not 3z. Additionally, step 3 incorrectly converts z to rho*cos(phi) but fails to square it (using rho instead of rho^2), leading to the wrong integrand.
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.