Divergence and curl
Problem 12.154 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle \cos{\left(z \right)}, - 3 y, x \rangle \).
- \[ \frac{d}{d z} x + \frac{d}{d y} \left(- 3 y\right) + \frac{d}{d x} \cos{\left(z \right)} = -3 \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} x - \frac{d}{d z} \left(- 3 y\right)\\- \frac{d}{d x} x + \frac{d}{d z} \cos{\left(z \right)}\\\frac{d}{d x} \left(- 3 y\right) - \frac{d}{d y} \cos{\left(z \right)}\end{matrix}\right] = \left[\begin{matrix}0\\- \sin{\left(z \right)} - 1\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = -3,\quad \nabla\times\mathbf{F} = \langle 0, - \sin{\left(z \right)} - 1, 0 \rangle \)
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: pass — The solution correctly computes the divergence and curl using standard formulas. The algebraic steps are verified, and the final results match the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the divergence and curl using standard formulas. The algebraic steps are verified, and the final results match the stated answer.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The curl calculation is incorrect. The y-component of the curl is calculated as -sin(z) - 1, but the correct calculation is Derivative(cos(z), z) -gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.