Divergence and curl
Problem 12.121 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 3 x y z, y, 2 z^{2} \rangle \).
- \[ \frac{d}{d y} y + \frac{d}{d z} 2 z^{2} + \frac{\partial}{\partial x} \left(- 3 x y z\right) = - 3 y z + 4 z + 1 \]div F.✓ Proved
- \[ \left[\begin{matrix}- \frac{d}{d z} y + \frac{d}{d y} 2 z^{2}\\- \frac{d}{d x} 2 z^{2} + \frac{\partial}{\partial z} \left(- 3 x y z\right)\\\frac{d}{d x} y - \frac{\partial}{\partial y} \left(- 3 x y z\right)\end{matrix}\right] = \left[\begin{matrix}0\\- 3 x y\\3 x z\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - 3 y z + 4 z + 1,\quad \nabla\times\mathbf{F} = \langle 0, - 3 x y, 3 x z \rangle \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field should result in a scalar and a 3D vector, respectively. The solution computes the correct divergence and curl components but fails to present them in the correct format or combines them incorrectly into a 4-element vector.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field should result in a scalar and a 3D vector, respectively. The solution computes the correct divergence and curl components but fails to present them in the correct format or combines them incorrectly into a 4-element vector.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field should result in a scalar and a 3D vector, respectively. The solution calculates the correct divergence and curl components but fails to format the final answer correctly, and the stated answer includes an erroneous extra component or formatting error.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.