Divergence and curl
Problem 12.100 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle z^{2}, 3 \sin{\left(y \right)}, z^{2} \rangle \).
- \[ \frac{d}{d x} z^{2} + \frac{d}{d z} z^{2} + \frac{d}{d y} 3 \sin{\left(y \right)} = 2 z + 3 \cos{\left(y \right)} \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} z^{2} - \frac{d}{d z} 3 \sin{\left(y \right)}\\- \frac{d}{d x} z^{2} + \frac{d}{d z} z^{2}\\- \frac{d}{d y} z^{2} + \frac{d}{d x} 3 \sin{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}0\\2 z\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = 2 z + 3 \cos{\left(y \right)},\quad \nabla\times\mathbf{F} = \langle 0, 2 z, 0 \rangle \)
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 is a scalar and the curl is a 3-vector. The solution computes them correctly but fails to present the final answer in the requested format or structure, resulting in a dimension mismatch.
Every verdict on record (4)
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 correctly but fails to present the final answer in the requested format or structure, resulting in a dimension mismatch.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer contains the divergence as the first component of a 4x1 matrix, which is structurally incorrect (divergence is a scalar). Additionally, the curl calculation in line 2 is incorrect: the y-component is calculated as -F_x_z + F_z_x = 0 + 0 = 0, but the solution claims it is 2*z. The correct curl is <0, 0, 0>.gpt-oss:20b: fail (misleading) 2026-09-26 — The divergence is computed correctly, but the reasoning mixes up the partial derivatives: it adds ∂/∂z of the first component instead of ∂/∂y of the second. This misstates the application of the divergence formula and could mislead a student.
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.