Home›Calculus 3›Divergence and curl›Problem 12.142

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 \( \nabla\cdot\mathbf{F} = 2 x - 2 y,\quad \nabla\times\mathbf{F} = \langle 1, 0, 2 x + 2 y \rangle \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify

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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04
gpt-oss:20b: pass 2026-10-04
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 2026-09-29
qwen3.6:27b-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 2026-10-04 with SymPy 1.14.0.