Divergence and curl
Problem 12.144 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle x y z, 2 x^{2}, y \rangle \).
- \[ \frac{d}{d y} 2 x^{2} + \frac{d}{d z} y + \frac{\partial}{\partial x} x y z = y z \]div F.✓ Proved
- \[ \left[\begin{matrix}- \frac{d}{d z} 2 x^{2} + \frac{d}{d y} y\\- \frac{d}{d x} y + \frac{\partial}{\partial z} x y z\\\frac{d}{d x} 2 x^{2} - \frac{\partial}{\partial y} x y z\end{matrix}\right] = \left[\begin{matrix}1\\x y\\- x z + 4 x\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = y z,\quad \nabla\times\mathbf{F} = \langle 1, x y, - x z + 4 x \rangle \)
✓ 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. How we verify
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, and the algebraic simplifications are verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the divergence and curl using standard formulas, and the algebraic simplifications are verified.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-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 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly sets up the formulas for divergence and curl and computes the derivatives accurately.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 2026-10-04 with SymPy 1.14.0.