Home›Calculus 3›Divergence and curl›Problem 12.73

Divergence and curl

Problem 12.73 · easy

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

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

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

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: fail (error) — The stated answer includes a fourth component `2*exp(x)`, which is not part of the curl (a vector field) or divergence (a scalar). The solution calculates the correct 3-component curl, but the stated answer is malformed and inconsistent with the solution steps.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer includes a fourth component `2*exp(x)`, which is not part of the curl (a vector field) or divergence (a scalar). The solution calculates the correct 3-component curl, but the stated answer is malformed and inconsistent with the solution steps.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer includes a fourth component `2*exp(x)`, which is not part of the curl (a vector in 3D has 3 components) nor the divergence (a scalar). The solution correctly computes the divergence and curl, but the stated answer is malformed and inconsistent with the computed results.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer includes a fourth component `2*exp(x)`, which is incorrect for the curl of a 3D vector field (which must have 3 components). The solution steps correctly calculate the 3-component curl, but the final stated answer is malformed and inconsistent with the steps.
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 2026-09-28 with SymPy 1.14.0.