Home›Calculus 3›Divergence and curl›Problem 12.71

Divergence and curl

Problem 12.71 · easy

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

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

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

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 of the given vector field. The algebraic steps are verified, and the final results match the stated answer.

Every verdict on record (6)

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: pass 2026-09-27 — The solution correctly computes the divergence and curl of the vector field. The intermediate steps are algebraically correct and match the stated answer.
gpt-oss:20b: fail (error) 2026-09-27 — The divergence calculation incorrectly differentiates the second component: it uses ∂(x y z)/∂z instead of ∂(2 x y z)/∂y, leading to an incorrect result.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer contains an extra row `[x*y + 2*x*z - 2]` which corresponds to the divergence, but the problem asks for divergence and curl as separate entities or the output format is malformed. More critically, the curl calculation in the solution is correct, but the stated answer matrix has 4 rows, implying a mix of divergence and curl or a formatting error that makes the answer incorrect as a standard vector/matrix representation of curl. The divergence is a scalar, not a vector component in the curl matrix. The provided 'Stated answer' is structurally invalid for the question asked (which usually expects two separate results or a specific combined format not shown here). However, looking closely at the solution steps: Step 1 computes divergence correctly. Step 2 computes curl correctly. The 'Stated answer' seems to concatenate the divergence (as a 1x1 matrix?) with the curl (3x1 matrix) into a 4x1 matrix. This is not a standard mathematical object for 'divergence and curl'. If the question implies returning them separately, the single matrix is wrong. If it implies a specific format, it's ambiguous. But usually, 'Find the divergence and curl' expects two answers. Providing a single 4x1 matrix is misleading/wrong format. Let's look at the curl components in the stated answer: `[-2*x*y + x*z]`, `[-y*z]`, `[2*y*z]`. These match the solution. The first row is the divergence. This is a non-standard concatenation. Is it an error? Yes, because divergence is a scalar field and curl is a vector field; they are not components of the same vector. Presenting them as a single column vector is mathematically incorrect.
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.