Home›Calculus 3›Divergence and curl at a point›Problem 12.392

Divergence and curl at a point

Problem 12.392 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, - 2 y z, x z^{2} \right\rangle \) at \( \displaystyle (1, 2, -1) \).

\[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} \left(- 2 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }} = 4 \]
div F = P_x + Q_y + R_z, at the point.✓ Proved
\[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} \left(- 2 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} \left(- 2 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\end{matrix}\right] = \left[\begin{matrix}4\\-1\\-1\end{matrix}\right] \]
curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved

Answer \( \nabla\cdot\mathbf F = 4,\quad \nabla\times\mathbf F = \left\langle 4, -1, -1 \right\rangle \)

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. 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	central differences of F at the point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The curl calculation in line 2 uses an incorrect formula. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>, but the solution com

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The curl calculation in line 2 uses an incorrect formula. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>, but the solution com
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The curl calculation is incorrect. The solution computes the components as (R_y - Q_z, P_z - R_x, Q_x - P_y), but the standard definition is (R_y - Q_z, P_x - R_z, Q_x - P_y). Specifically, the y-component should be P_x - R_z, not P_z - R_x.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The curl calculation is incorrect. The solution uses the wrong components for the curl formula (e.g., using R_y instead of R_x for the first component, and mixing up signs and variables for the others). The correct curl is <2xz, -z^2, 2xy> which evaluates to <-2, -1, 4> at the point, not <4, -1, -1>.
gpt-oss:20b: pass 2026-10-14

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/div_curl_point, checked 2026-10-18 with SymPy 1.14.0.