Home›Calculus 3›Divergence and curl›Problem 12.69

Divergence and curl

Problem 12.69 · easy

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

\[ \frac{d}{d x} 2 x + \frac{d}{d y} \left(- z\right) + \frac{d}{d z} 2 \cos{\left(z \right)} = 2 - 2 \sin{\left(z \right)} \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- z\right) + \frac{d}{d y} 2 \cos{\left(z \right)}\\\frac{d}{d z} 2 x - \frac{d}{d x} 2 \cos{\left(z \right)}\\- \frac{d}{d y} 2 x + \frac{d}{d x} \left(- z\right)\end{matrix}\right] = \left[\begin{matrix}1\\0\\0\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = 2 - 2 \sin{\left(z \right)},\quad \nabla\times\mathbf{F} = \langle 1, 0, 0 \rangle \)

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. 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 — The solution correctly computes the divergence and curl using standard formulas. The algebraic steps are verified, and the final results match the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the divergence and curl using standard formulas. The algebraic steps are verified, and the final results match the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the divergence and curl of the given vector field. The intermediate steps are algebraically verified, and the final results match the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the curl as <1, 0, 0>, but the final stated answer format is incorrect and inconsistent with the computed result.
gpt-oss:20b: pass 2026-09-26

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 2026-09-28 with SymPy 1.14.0.