Home›Calculus 3›Divergence and curl›Problem 12.103

Divergence and curl

Problem 12.103 · easy

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

\[ \frac{d}{d y} \left(- 2 x\right) + \frac{d}{d x} 2 z^{2} + \frac{d}{d z} 3 \sin{\left(y \right)} = 0 \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- 2 x\right) + \frac{d}{d y} 3 \sin{\left(y \right)}\\\frac{d}{d z} 2 z^{2} - \frac{d}{d x} 3 \sin{\left(y \right)}\\\frac{d}{d x} \left(- 2 x\right) - \frac{d}{d y} 2 z^{2}\end{matrix}\right] = \left[\begin{matrix}3 \cos{\left(y \right)}\\4 z\\-2\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = 0,\quad \nabla\times\mathbf{F} = \langle 3 \cos{\left(y \right)}, 4 z, -2 \rangle \)

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: pass — The solution correctly applies the definitions of divergence and curl to the given vector field. The algebraic simplifications are correct, and the final matrix matches the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the definitions of divergence and curl to the given vector field. The algebraic simplifications are correct, and the final matrix matches the stated answer.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution computes them separately but fails to combine them into the requested format or explain the discrepancy. Additionally, the divergence calculation line is syntactically confusing (summing derivatives but equating to 0 without showing the sum is 0).
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 2026-09-26 with SymPy 1.14.0.