Home›Calculus 3›Divergence and curl›Problem 12.127

Divergence and curl

Problem 12.127 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 3 \sin{\left(y \right)}, - x y z, y^{2} \rangle \).

\[ \frac{d}{d z} y^{2} + \frac{\partial}{\partial y} \left(- x y z\right) + \frac{d}{d x} 3 \sin{\left(y \right)} = - x z \]
div F.✓ Proved
\[ \left[\begin{matrix}\frac{d}{d y} y^{2} - \frac{\partial}{\partial z} \left(- x y z\right)\\- \frac{d}{d x} y^{2} + \frac{d}{d z} 3 \sin{\left(y \right)}\\\frac{\partial}{\partial x} \left(- x y z\right) - \frac{d}{d y} 3 \sin{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}x y + 2 y\\0\\- y z - 3 \cos{\left(y \right)}\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = - x z,\quad \nabla\times\mathbf{F} = \langle x y + 2 y, 0, - y z - 3 \cos{\left(y \right)} \rangle \)

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 vector field. The intermediate symbolic expressions are correct, and the final simplified 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 vector field. The intermediate symbolic expressions are correct, and the final simplified results match the stated answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field should result in a scalar and a 3D vector, respectively. The solution computes the correct components but fails to assemble them into the correct final format or explicitly state the divergence as a scalar separate from the curl vector.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence and curl of a 3D vector field are a scalar and a 3D vector, respectively. The solution computes the correct components but fails to present them in the correct format or combine them into a single coherent answer as implied by the problem statement.
gpt-oss:20b: pass 2026-09-27

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.