Home›Calculus 3›Divergence and curl›Problem 12.120

Divergence and curl

Problem 12.120 · easy

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

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

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

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 is a 4x1 matrix, but the divergence and curl of a 3D vector field are a scalar and a 3D vector, respectively. The solution correctly computes the scalar divergence and 3D curl vector, but fails to present them in the format requested by the stated answer (or the stated answer is malformed). Furthermore, the solution does not explicitly state the final combined result matching the 'Stated answer' format, making it unclear how the scalar and vector are combined into the provided 4x1 matrix structure.

Every verdict on record (4)

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 correctly computes the scalar divergence and 3D curl vector, but fails to present them in the format requested by the stated answer (or the stated answer is malformed). Furthermore, the solution does not explicitly state the final combined result matching the 'Stated answer' format, making it unclear how the scalar and vector are combined into the provided 4x1 matrix structure.
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 them correctly but fails to present them in the expected format or combine them correctly.
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 2026-09-27 with SymPy 1.14.0.