Home›Calculus 3›Divergence and curl›Problem 12.126

Divergence and curl

Problem 12.126 · easy

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

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

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

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.

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 is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution correctly computes them separately but fails to explain how they combine into the provided matrix format, which appears to be a concatenation of [div, curl] with an extra zero or a formatting error in the problem statement's expected output. More critically, the first line claims the divergence is `x*y + 4*x + 2*y`, which is incorrect; the divergence of <2x^2, y^2, xyz> is 4x + 2y + xy. Wait, the equation line says `Derivative(2*x**2, x) + Derivative(y**2, y) + Derivative(x*y*z, z)` equals `x*y + 4*x + 2*y`. Let's check: d/dx(2x^2)=4x, d/dy(y^2)=2y, d/dz(xyz)=xy. Sum is 4x+2y+xy. The equation is marked 'proved' by a CAS, so the algebra is correct. However, the 'Stated answer' matrix has `x*y + 4*x + 2*y` as the first element, then `x*z`, `-y*z`, `0`. This looks like [div, curl_x, curl_y, curl_z]. The solution calculates div and curl correctly. The issue is that the solution does not explicitly state that the final answer is the concatenation of the divergence and curl, or that the problem asks for a specific vector format. But looking closer at the 'Stated answer', it is a 4x1 matrix. The solution provides two separate equations. It does not combine them. Is this an error? The prompt asks to review the solution. The solution lines are correct. The 'Stated answer' is just the target. The solution derives the components. The only potential issue is if the student doesn't know how to assemble the final answer. But the lines themselves are correct. Let's re-read the divergence calculation. `x*y + 4*x + 2*y`. Correct. Curl: `xz`, `-yz`, `0`. Correct. The solution is mathematically sound. Why would it fail? Perhaps the 'Stated answer' implies a specific ordering or format that isn't explained? Or is there a subtle error? Let's check the curl again. Curl F = (dFz/dy - dFy/dz, dFx/dz - dFz/dx, dFy/dx - dFx/dy). F = (2x^2, y^2, xyz). dFz/dy = xz. dFy/dz = 0. Comp 1: xz. dFx/dz = 0. dFz/dx = yz. Comp 2: -yz. dFy/dx = 0. dFx/dy = 0. Comp 3: 0. Result <xz, -yz, 0>. Correct. The solution is correct. I will mark it pass.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution correctly computes them separately but fails to explain how they combine into the provided matrix format, which appears to be a concatenation of [div, curl] with an extra zero or a formatting error in the problem statement's expected output. More critically, the first line claims the divergence is `x*y + 4*x + 2*y`, which is incorrect; the divergence of <2x^2, y^2, xyz> is 4x + 2y + xy. Wait, the equation line says `Derivative(2*x**2, x) + Derivative(y**2, y) + Derivative(x*y*z, z)` equals `x*y + 4*x + 2*y`. Let's check: d/dx(2x^2)=4x, d/dy(y^2)=2y, d/dz(xyz)=xy. Sum is 4x+2y+xy. The equation is marked 'proved' by a CAS, so the algebra is correct. However, the 'Stated answer' matrix has `x*y + 4*x + 2*y` as the first element, then `x*z`, `-y*z`, `0`. This looks like [div, curl_x, curl_y, curl_z]. The solution calculates div and curl correctly. The issue is that the solution does not explicitly state that the final answer is the concatenation of the divergence and curl, or that the problem asks for a specific vector format. But looking closer at the 'Stated answer', it is a 4x1 matrix. The solution provides two separate equations. It does not combine them. Is this an error? The prompt asks to review the solution. The solution lines are correct. The 'Stated answer' is just the target. The solution derives the components. The only potential issue is if the student doesn't know how to assemble the final answer. But the lines themselves are correct. Let's re-read the divergence calculation. `x*y + 4*x + 2*y`. Correct. Curl: `xz`, `-yz`, `0`. Correct. The solution is mathematically sound. Why would it fail? Perhaps the 'Stated answer' implies a specific ordering or format that isn't explained? Or is there a subtle error? Let's check the curl again. Curl F = (dFz/dy - dFy/dz, dFx/dz - dFz/dx, dFy/dx - dFx/dy). F = (2x^2, y^2, xyz). dFz/dy = xz. dFy/dz = 0. Comp 1: xz. dFx/dz = 0. dFz/dx = yz. Comp 2: -yz. dFy/dx = 0. dFx/dy = 0. Comp 3: 0. Result <xz, -yz, 0>. Correct. The solution is correct. I will mark it pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution calculates the divergence and curl separa
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.