Divergence and curl
Problem 12.142 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x y, 2 x y, y \rangle \).
- \[ \frac{d}{d z} y + \frac{\partial}{\partial x} \left(- 2 x y\right) + \frac{\partial}{\partial y} 2 x y = 2 x - 2 y \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} y - \frac{\partial}{\partial z} 2 x y\\- \frac{d}{d x} y + \frac{\partial}{\partial z} \left(- 2 x y\right)\\- \frac{\partial}{\partial y} \left(- 2 x y\right) + \frac{\partial}{\partial x} 2 x y\end{matrix}\right] = \left[\begin{matrix}1\\0\\2 x + 2 y\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = 2 x - 2 y,\quad \nabla\times\mathbf{F} = \langle 1, 0, 2 x + 2 y \rangle \)
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 3x1 vector respectively. The solution calculates the correct components but fails to format the final answer correctly as a combined object or separate results matching the problem's likely expectation (usually two separate answers or a specific structure). More critically, the 'Stated answer' provided in the prompt is structurally invalid for the question asked (mixing a scalar and a vector into a 4x1 matrix without context), and the solution steps do not explicitly construct this malformed final output, making the connection between the steps and the stated answer unclear or erroneous.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field should result in a scalar and a 3x1 vector respectively. The solution calculates the correct components but fails to format the final answer correctly as a combined object or separate results matching the problem's likely expectation (usually two separate answers or a specific structure). More critically, the 'Stated answer' provided in the prompt is structurally invalid for the question asked (mixing a scalar and a vector into a 4x1 matrix without context), and the solution steps do not explicitly construct this malformed final output, making the connection between the steps and the stated answer unclear or erroneous.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3-component curl in line 2, but the final answer format is inconsistent and incorrect.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.