Home›Calculus 3›Divergence and curl›Problem 12.98

Divergence and curl

Problem 12.98 · easy

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

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

Answer \( \nabla\cdot\mathbf{F} = 2 x y,\quad \nabla\times\mathbf{F} = \langle 2 x z, - 2 y z - y, z - 3 e^{x} \rangle \)

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.

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 computes them correctly but fails to combine them into the requested format or explain the discrepancy.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer is a 4x1 matrix, but the divergence is a scalar and the curl is a 3-vector. The solution computes them correctly but fails to combine them into the requested format or explain the discrepancy.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The stated answer is a 4x1 matrix, but the divergence and curl of a vector field are a scalar and a 3-vector, respectively. The solution computes them separately but fails to combine them into the correct output format implied by the answer (likely a column vector containing the divergence followed by the curl components, or vice versa, but the dimensions don't match a standard single object without clarification). More critically, the stated answer has 4 entries: [2xy, 2xz, -2yz-y, z-3exp(x)]. The divergence is 2xy. The curl is <2xz, -2yz-y, z-3exp(x)>. The stated answer lists the divergence as the first component and the curl components as the next three. However, the solution lines only show the computation of div and curl separately. The 'Stated answer' format is ambiguous or non-standard for 'Find divergence and curl' (usually two separate answers). If the problem implies a single vector output, the solution does not explicitly construct it. But looking closer, the main issue is likely that the solution presents two separate results while the 'Stated answer' is a single matrix. The review instructions say 'Stated answer' is the target. The solution lines 1 and 2 compute the correct values. Line 1 computes div = 2xy. Line 2 computes curl = <2xz, -2yz-y, z-3exp(x)>. The stated answer is a column vector [2xy, 2xz, -2yz-y, z-3exp(x)]^T. This looks like [div, curl_x, curl_y, curl_z]^T. The solution does not explicitly state this combination. However, usually 'Find divergence and curl' expects two separate answers. If the 'Stated answer' is the ground truth format, the solution is incomplete because it doesn't assemble the final object. But is it 'wrong'? It computes the correct components. Let's look for a calculation error. Div: d/dx(-yz) + d/dy(-3e^x) + d/dz(2xyz) = 0 + 0 + 2xy = 2xy. Correct. Curl: i(d/dy(2xyz)-d/dz(-3e^x)) - j(d/dx(2xyz)-d/dz(-yz)) + k(d/dx(-3e^x)-d/dy(-yz)). i(2xz - 0) - j(2yz - (-y)) + k(-3e^x - (-z)) = <2xz, -2yz-y, z-3e^x>. Correct. The solution is mathematically correct. The 'Stated answer' format is just a concatenation. The solution lines are correct. Why would it fail? Maybe the prompt implies the solution *is* the text provided. The text provided is just the equations. It doesn't explicitly say 'The divergence is ... and the curl is ...'. But it labels them. Is there a subtle error? No. Wait, look at the stated answer again. It is a 4x1 matrix. The problem asks for divergence AND curl. Usually these are separate. If the system expects a single vector, the solution should probably output that. But the solution lines are marked 'unchecked' equations. The review is of the *sentences* or the *logic*. There are no sentences. Just equations. The equations are correct. The verdict should be pass if the math is right. But wait, is the order of curl components correct? Standard curl is (F_z_y - F_y_z, F_x_z - F_z_x, F_y_x - F_x_y). Line 2: Row 1: d/dy(F_z) - d/dz(F_y) = 2xz - 0. Correct. Row 2: d/dz(F_x) - d/dx(F_z) = -y - 2yz. Correct. Row 3: d/dx(F_y) - d/dy(F_x) = -3e^x - (-z) = z - 3e^x. Correct. The math is perfect. The only potential issue is the format mismatch between 'two answers' and 'one matrix'. However, without a sentence explaining the concatenation, it's not 'wrong', just potentially incomplete in presentation. But the instructions say 'find what is wrong'. If nothing is wrong, pass. Let's look really closely. Is it possible the stated answer is wrong? No, I verified the math. Is the solution misleading? No. I will pass.
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.