Divergence and curl
Problem 12.141 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x^{2}, - 3 y z, \cos{\left(z \right)} \rangle \).
- \[ \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{\partial}{\partial y} \left(- 3 y z\right) + \frac{d}{d z} \cos{\left(z \right)} = - 4 x - 3 z - \sin{\left(z \right)} \]div F.✓ Proved
- \[ \left[\begin{matrix}- \frac{\partial}{\partial z} \left(- 3 y z\right) + \frac{d}{d y} \cos{\left(z \right)}\\\frac{d}{d z} \left(- 2 x^{2}\right) - \frac{d}{d x} \cos{\left(z \right)}\\- \frac{d}{d y} \left(- 2 x^{2}\right) + \frac{\partial}{\partial x} \left(- 3 y z\right)\end{matrix}\right] = \left[\begin{matrix}3 y\\0\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - 4 x - 3 z - \sin{\left(z \right)},\quad \nabla\times\mathbf{F} = \langle 3 y, 0, 0 \rangle \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 results are verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl using standard formulas. The algebraic results are verified as correct.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl of the given vector field. The algebraic steps are verified, and the final results match the stated answer.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.