Divergence and curl
Problem 12.101 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - x^{2}, 2 \sin{\left(y \right)}, e^{x} \rangle \).
- \[ \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d z} e^{x} + \frac{d}{d y} 2 \sin{\left(y \right)} = - 2 x + 2 \cos{\left(y \right)} \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} e^{x} - \frac{d}{d z} 2 \sin{\left(y \right)}\\\frac{d}{d z} \left(- x^{2}\right) - \frac{d}{d x} e^{x}\\- \frac{d}{d y} \left(- x^{2}\right) + \frac{d}{d x} 2 \sin{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}0\\- e^{x}\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - 2 x + 2 \cos{\left(y \right)},\quad \nabla\times\mathbf{F} = \langle 0, - e^{x}, 0 \rangle \)
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 computes the correct components but fails to present them in the correct format or combine them as requested.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — 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 the correct components but fails to present them in the correct format or combine them as requested.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-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 correctly computes the components but fails to format the final answer as requested or implied by the standard definitions, resulting in a dimension mismatch in the stated answer.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.