Home›Calculus 3›Divergence and curl›Problem 12.99

Divergence and curl

Problem 12.99 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle 2 x z, 3 \cos{\left(z \right)}, 2 y^{2} \rangle \).

\[ \frac{d}{d z} 2 y^{2} + \frac{\partial}{\partial x} 2 x z + \frac{d}{d y} 3 \cos{\left(z \right)} = 2 z \]
div F.✓ Proved
\[ \left[\begin{matrix}\frac{d}{d y} 2 y^{2} - \frac{d}{d z} 3 \cos{\left(z \right)}\\- \frac{d}{d x} 2 y^{2} + \frac{\partial}{\partial z} 2 x z\\- \frac{\partial}{\partial y} 2 x z + \frac{d}{d x} 3 \cos{\left(z \right)}\end{matrix}\right] = \left[\begin{matrix}4 y + 3 \sin{\left(z \right)}\\2 x\\0\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = 2 z,\quad \nabla\times\mathbf{F} = \langle 4 y + 3 \sin{\left(z \right)}, 2 x, 0 \rangle \)

✓ 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 computes the divergence and curl using standard formulas. The algebraic results are verified as correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the divergence and curl using standard formulas. The algebraic results are verified as correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the divergence and curl of the given vector field. The intermediate steps and final results are mathematically sound.
gpt-oss:20b: pass 2026-09-26

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 2026-09-26 with SymPy 1.14.0.