Home›Calculus 3›Divergence and curl›Problem 12.159

Divergence and curl

Problem 12.159 · easy

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

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

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

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the definitions of divergence and curl to the given vector field. The algebraic results match the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the definitions of divergence and curl to the given vector field. The algebraic results match the stated answer.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the divergence and curl using the standard definitions. The algebraic results match the stated answer.
gpt-oss:20b: pass 2026-10-03

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.