Home›Calculus 3›Divergence and curl›Problem 12.75

Divergence and curl

Problem 12.75 · easy

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

\[ \frac{d}{d x} \left(- x\right) + \frac{d}{d z} 2 y^{2} + \frac{\partial}{\partial y} \left(- x z\right) = -1 \]
div F.✓ Proved
\[ \left[\begin{matrix}\frac{d}{d y} 2 y^{2} - \frac{\partial}{\partial z} \left(- x z\right)\\\frac{d}{d z} \left(- x\right) - \frac{d}{d x} 2 y^{2}\\- \frac{d}{d y} \left(- x\right) + \frac{\partial}{\partial x} \left(- x z\right)\end{matrix}\right] = \left[\begin{matrix}x + 4 y\\0\\- z\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = -1,\quad \nabla\times\mathbf{F} = \langle x + 4 y, 0, - z \rangle \)

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. 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 of the given vector field. 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 of the given vector field. 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 algebraic steps are 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 3D curl in line 2, but the stated answer includes an erroneous fourth component (likely a formatting artifact or confusion with a 4D problem).
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.