Divergence and curl
Problem 12.122 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 3 \cos{\left(z \right)}, - \sin{\left(y \right)}, x^{2} \rangle \).
- \[ \frac{d}{d z} x^{2} + \frac{d}{d y} \left(- \sin{\left(y \right)}\right) + \frac{d}{d x} 3 \cos{\left(z \right)} = - \cos{\left(y \right)} \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} x^{2} - \frac{d}{d z} \left(- \sin{\left(y \right)}\right)\\- \frac{d}{d x} x^{2} + \frac{d}{d z} 3 \cos{\left(z \right)}\\\frac{d}{d x} \left(- \sin{\left(y \right)}\right) - \frac{d}{d y} 3 \cos{\left(z \right)}\end{matrix}\right] = \left[\begin{matrix}0\\- 2 x - 3 \sin{\left(z \right)}\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - \cos{\left(y \right)},\quad \nabla\times\mathbf{F} = \langle 0, - 2 x - 3 \sin{\left(z \right)}, 0 \rangle \)
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.
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: passqwen3.6:27b-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 calculates them correctly but fails to present the final answer in the correct format or combine them properly.
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 calculates them correctly but fails to present the final answer in the correct format or combine them properly.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-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 fails to combine the results into the correct format or even correctly identify the scalar divergence (it lists it as part of a tuple in line 1).gpt-oss:20b: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The divergence calculation incorrectly differentiates \(x^2\) with respect to \(z\) and \(3\cos z\) with respect to \(x\); the correct divergence is \(\pa
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.