Home›Calculus 3›Divergence and curl at a point›Problem 12.474

Divergence and curl at a point

Problem 12.474 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, - y z, x z^{2} \right\rangle \) at \( \displaystyle (0, 1, 2) \).

\[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} - y z \right|_{\substack{ x=0\\ y=1\\ z=2 }} = -2 \]
div F = P_x + Q_y + R_z, at the point.✓ Proved
\[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} - y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} - y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\end{matrix}\right] = \left[\begin{matrix}1\\-4\\0\end{matrix}\right] \]
curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved

Answer \( \nabla\cdot\mathbf F = -2,\quad \nabla\times\mathbf F = \left\langle 1, -4, 0 \right\rangle \)

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: pass — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The curl calculation in line 2 is incorrect. The first component should be R_y - Q_z = 0 - (-x) = x, which evaluates to 0 at (0,1,2), not 1. The second component should be P_z - R_x = 0 - z^2 = -4, which is correct. The third component Q_x - P_y = 0 - x^2 = 0, which is correct. The stated answer for the curl is wrong.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.