Home›Calculus 3›Divergence and curl›Problem 12.134

Divergence and curl

Problem 12.134 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x, 2 y^{2}, x \rangle \).

\[ \frac{d}{d z} x + \frac{d}{d x} 2 x + \frac{d}{d y} 2 y^{2} = 4 y + 2 \]
div F.✓ Proved
\[ \left[\begin{matrix}\frac{d}{d y} x - \frac{d}{d z} 2 y^{2}\\- \frac{d}{d x} x + \frac{d}{d z} 2 x\\- \frac{d}{d y} 2 x + \frac{d}{d x} 2 y^{2}\end{matrix}\right] = \left[\begin{matrix}0\\-1\\0\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = 4 y + 2,\quad \nabla\times\mathbf{F} = \langle 0, -1, 0 \rangle \)

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. 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the divergence and curl using standard formulas, and the algebraic results are verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the divergence and curl using standard formulas, and the algebraic results are verified.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — 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 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer includes a fourth component (0), implying a 4D vector, but the curl of a 3D vector field is a 3D vector. The solution correctly computes the 3D curl, but the stated answer format is inconsistent with standard vector calculus definitions for 3D fields.
gpt-oss:20b: pass 2026-09-28

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 2026-10-03 with SymPy 1.14.0.