Divergence and curl
Problem 12.138 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle y z, - 3 y z, 3 e^{x} \rangle \).
- \[ \frac{\partial}{\partial y} \left(- 3 y z\right) + \frac{\partial}{\partial x} y z + \frac{d}{d z} 3 e^{x} = - 3 z \]div F.✓ Proved
- \[ \left[\begin{matrix}- \frac{\partial}{\partial z} \left(- 3 y z\right) + \frac{d}{d y} 3 e^{x}\\\frac{\partial}{\partial z} y z - \frac{d}{d x} 3 e^{x}\\\frac{\partial}{\partial x} \left(- 3 y z\right) - \frac{\partial}{\partial y} y z\end{matrix}\right] = \left[\begin{matrix}3 y\\y - 3 e^{x}\\- z\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - 3 z,\quad \nabla\times\mathbf{F} = \langle 3 y, y - 3 e^{x}, - z \rangle \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ 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 | difference quotients of each component at (0.3, 0.7, 1.1) agree |
Reviewers
gpt-oss:20b: fail (error) — The divergence is computed with the wrong partial derivatives: it should be ∂(yz)/∂x + ∂(-3yz)/∂y + ∂(3e^x)/∂z = z - 3z + 0 = -2z, not -3z. The curl components are also mis‑ordered and mis‑calculated, leading to an incorrect result.qwen3.6:27b-mlx: pass — The solution correctly computes the divergence and curl of the given vector field. The algebraic steps are verified, and the final results match the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl of the given vector field. The algebraic steps are verified, and the final results match the stated answer.gpt-oss:20b: fail (error) 2026-09-29 — The divergence is computed with the wrong partial derivatives: it should be ∂(yz)/∂x + ∂(-3yz)/∂y + ∂(3e^x)/∂z = z - 3z + 0 = -2z, not -3z. The curl components are also mis‑ordered and mis‑calculated, leading to an incorrect result.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl of the given vector field. The algebraic steps are verified, and the final result matches the stated answer.gpt-oss:20b: pass 2026-09-29
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_curl, checked 2026-09-29 with SymPy 1.14.0.