Divergence and curl
Problem 12.143 · easy
Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 e^{x}, - 2 y z, 2 y \rangle \).
- \[ \frac{d}{d z} 2 y + \frac{\partial}{\partial y} \left(- 2 y z\right) + \frac{d}{d x} \left(- 2 e^{x}\right) = - 2 z - 2 e^{x} \]div F.✓ Proved
- \[ \left[\begin{matrix}\frac{d}{d y} 2 y - \frac{\partial}{\partial z} \left(- 2 y z\right)\\- \frac{d}{d x} 2 y + \frac{d}{d z} \left(- 2 e^{x}\right)\\\frac{\partial}{\partial x} \left(- 2 y z\right) - \frac{d}{d y} \left(- 2 e^{x}\right)\end{matrix}\right] = \left[\begin{matrix}2 y + 2\\0\\0\end{matrix}\right] \]curl F, component by component.✓ Proved
Answer \( \nabla\cdot\mathbf{F} = - 2 z - 2 e^{x},\quad \nabla\times\mathbf{F} = \langle 2 y + 2, 0, 0 \rangle \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes a 3x1 matrix for the curl, but the stated answer includes an extra zero row, making it dimensionally inconsistent with the standard definition of curl.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The stated answer is a 4x1 matrix, but the curl of a 3D vector field is a 3D vector. The solution correctly computes a 3x1 matrix for the curl, but the stated answer includes an extra zero row, making it dimensionally inconsistent with the standard definition of curl.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — 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 separately but fails to combine them into the requested format or explain the discrepancy. Additionally, the curl calculation in line 2 is incorrect: the y-component of curl F is \partial_x F_z - \partial_z F_x = 0 - 0 = 0, but the solution shows 2*y + 2 (which looks like a mix-up with the divergence or a different component). Wait, let's re-read carefully. Line 2 computes curl components. \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2+2y. This is the x-component. The solution labels this matrix as curl F. The stated answer has 4 rows. This is a format error in the final answer relative to standard vector calculus outputs, but more importantly, the solution does not produce the stated answer. The stated answer seems to concatenate divergence and curl? Div = -2z - 2e^x. Curl = <2+2y, 0, 0>. The stated answer is [-2z - 2e^x, 2y+2, 0, 0]^T. This looks like [Div, Curl_x, Curl_y, Curl_z]^T. However, the solution text does not state this concatenation rule. It just calculates div and curl separately. The prompt asks to review the solution. The solution calculates div and curl. The stated answer is a specific matrix. Does the solution justify the stated answer? No. It just lists two separate calculations. Furthermore, is the curl calculation correct? Curl F = \nabla \times F. x-comp: \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2+2y. y-comp: \partial_z(-2e^x) - \partial_x(2y) = 0 - 0 = 0. z-comp: \partial_x(-2yz) - \partial_y(-2e^x) = 0 - 0 = 0. So Curl is <2+2y, 0, 0>. Div is -2e^x - 2z. The stated answer combines them. The solution does not explain this combination. This is misleading/incomplete. Also, the solution marks the equations as 'unchecked' but the prompt says 'computer algebra system has already checked... do not re-check'. I should focus on the sentences/logic. The logic of 'here is div, here is curl' does not lead to the single matrix answer provided without an explicit step saying 'we stack them'. This is a missing step/logic gap.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.