Home›Calculus 3›Divergence and curl›Problem 12.140

Divergence and curl

Problem 12.140 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x^{2}, - 2 x, x y z \rangle \).

\[ \frac{d}{d y} \left(- 2 x\right) + \frac{d}{d x} 2 x^{2} + \frac{\partial}{\partial z} x y z = x y + 4 x \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- 2 x\right) + \frac{\partial}{\partial y} x y z\\\frac{d}{d z} 2 x^{2} - \frac{\partial}{\partial x} x y z\\\frac{d}{d x} \left(- 2 x\right) - \frac{d}{d y} 2 x^{2}\end{matrix}\right] = \left[\begin{matrix}x z\\- y z\\-2\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = x y + 4 x,\quad \nabla\times\mathbf{F} = \langle x z, - y z, -2 \rangle \)

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: pass
qwen3.6:27b-mlx: fail (error) — The stated answer includes a fourth component (the divergence) in a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the curl as a 3x1 matrix in step 2, but the final stated answer format is inconsistent with standard vector calculus definitions and mixes dimensions.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer includes a fourth component (the divergence) in a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the curl as a 3x1 matrix in step 2, but the final stated answer format is inconsistent with standard vector calculus definitions and mixes dimensions.
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 divergence is a scalar and the curl is a 3-vector. The solution fails to combine these into a single coherent object or explain the format, resulting in a dimensionally inconsistent final 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.