Home›Calculus 3›Divergence and curl›Problem 12.143

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. 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 curl calculation is incorrect. The x-component should be \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2 + 2y, but the solution claims the result is 2y + 2 (which is actually correct algebraically, wait). Let's re-read carefully. F = <-2e^x, -2yz, 2y>. Curl F = < \partial_y F_z - \partial_z F_y, \partial_z F_x - \partial_x F_z, \partial_x F_y - \partial_y F_x >. x-comp: \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2 + 2y. The solution says: Matrix([[Derivative(2*y, y) - Derivative(-2*y*z, z), ...]]) = Matrix([[2*y + 2], ...]). Derivative(2*y, y) is 2. Derivative(-2*y*z, z) is -2y. So 2 - (-2y) = 2 + 2y. The equation line is marked 'proved' by a CAS, so the algebra is correct. Wait, let's look at the stated answer in the problem description vs the solution. Stated answer: \nabla\times\mathbf{F} = \langle 2 y + 2, 0, 0 \rangle. Solution line 2 result: Matrix([[2*y + 2], [0], [0]]). Is the divergence correct? Div F = \partial_x(-2e^x) + \partial_y(-2yz) + \partial_z(2y) = -2e^x - 2z + 0 = -2e^x - 2z. Solution line 1: -2*z - 2*exp(x). Correct. So where is the error? Let's re-read the prompt's instructions. "Each line... is either an EQUATION... or a SENTENCE". The solution provided ONLY has equations. There are no sentences to review for setup, theorems, etc. However, usually, these tasks imply checking if the *equations* themselves represent the correct mathematical steps. The prompt says "do not re-check the algebra of a line marked proved". If the algebra is proved correct by CAS, and the setup (the derivatives chosen) matches the definition of div/curl, then it should pass. Let's double check the curl components in the matrix. Row 1: Derivative(2*y, y) - Derivative(-2*y*z, z). This corresponds to \partial_y F_z - \partial_z F_y. Correct for x-component. Row 2: -Derivative(2*y, x) + Derivative(-2*exp(x), z). This corresponds to -(\partial_x F_z) + (\partial_z F_x). Wait. The standard formula for y-component is \partial_z F_x - \partial_x F_z. F_x = -2e^x. \partial_z F_x = 0. F_z = 2y. \partial_x F_z = 0. So y-comp is 0 - 0 = 0. The equation has -Derivative(2*y, x) + Derivative(-2*exp(x), z). Derivative(2*y, x) is 0. Derivative(-2*exp(x), z) is 0. Result 0. Correct. Row 3: Derivative(-2*y*z, x) - Derivative(-2*exp(x), y). This corresponds to \partial_x F_y - \partial_y F_x. Correct for z-component. F_y = -2yz. \partial_x F_y = 0. F_x = -2e^x. \partial_y F_x = 0. Result 0. Correct. The solution seems mathematically sound and the algebra is verified. Why would it fail? Maybe the prompt implies I should check if the *sentences* are missing? "The sentences are what you are here for." But there are no sentences. If there are no sentences, can I fail it? Or is it a pass because the equations are correct? Usually, if the solution is just a list of verified equations that correctly solve the problem, it passes. The prompt says "Check that... each conclusion follows from the lines before it". Here, the equations *are* the conclusions. They follow from the definitions. Let's look closer at the divergence equation. Derivative(2*y, z) + Derivative(-2*y*z, y) + Derivative(-2*exp(x), x) \partial_z F_z + \partial_y F_y + \partial_x F_x. F_z = 2y. \partial_z(2y) = 0. F_y = -2yz. \partial_y(-2yz) = -2z. F_x = -2e^x. \partial_x(-2e^x) = -2e^x. Sum: -2z - 2e^x. Equation says: -2*z - 2*exp(x). Correct. Is there a subtle error in the curl matrix setup? Standard Curl: ( \partial_y F_z - \partial_z F_y ) i ( \partial_z F_x - \partial_x F_z ) j ( \partial_x F_y - \partial_y F_x ) k Solution Line 2: Row 1: Derivative(2*y, y) - Derivative(-2*y*z, z). \partial_y F_z - \partial_z F_y. Correct. Row 2: -Derivative(2*y, x) + Derivative(-2*exp(x), z). This is -\partial_x F_z + \partial_z F_x. Which is \partial_z F_x - \partial_x F_z. Correct. Row 3: Derivative(-2*y*z, x) - Derivative(-2*exp(x), y). \partial_x F_y - \partial_y F_x. Correct. Everything seems correct. The verdict should be pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The curl calculation is incorrect. The x-component should be \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2 + 2y, but the solution claims the result is 2y + 2 (which is actually correct algebraically, wait). Let's re-read carefully. F = <-2e^x, -2yz, 2y>. Curl F = < \partial_y F_z - \partial_z F_y, \partial_z F_x - \partial_x F_z, \partial_x F_y - \partial_y F_x >. x-comp: \partial_y(2y) - \partial_z(-2yz) = 2 - (-2y) = 2 + 2y. The solution says: Matrix([[Derivative(2*y, y) - Derivative(-2*y*z, z), ...]]) = Matrix([[2*y + 2], ...]). Derivative(2*y, y) is 2. Derivative(-2*y*z, z) is -2y. So 2 - (-2y) = 2 + 2y. The equation line is marked 'proved' by a CAS, so the algebra is correct. Wait, let's look at the stated answer in the problem description vs the solution. Stated answer: \nabla\times\mathbf{F} = \langle 2 y + 2, 0, 0 \rangle. Solution line 2 result: Matrix([[2*y + 2], [0], [0]]). Is the divergence correct? Div F = \partial_x(-2e^x) + \partial_y(-2yz) + \partial_z(2y) = -2e^x - 2z + 0 = -2e^x - 2z. Solution line 1: -2*z - 2*exp(x). Correct. So where is the error? Let's re-read the prompt's instructions. "Each line... is either an EQUATION... or a SENTENCE". The solution provided ONLY has equations. There are no sentences to review for setup, theorems, etc. However, usually, these tasks imply checking if the *equations* themselves represent the correct mathematical steps. The prompt says "do not re-check the algebra of a line marked proved". If the algebra is proved correct by CAS, and the setup (the derivatives chosen) matches the definition of div/curl, then it should pass. Let's double check the curl components in the matrix. Row 1: Derivative(2*y, y) - Derivative(-2*y*z, z). This corresponds to \partial_y F_z - \partial_z F_y. Correct for x-component. Row 2: -Derivative(2*y, x) + Derivative(-2*exp(x), z). This corresponds to -(\partial_x F_z) + (\partial_z F_x). Wait. The standard formula for y-component is \partial_z F_x - \partial_x F_z. F_x = -2e^x. \partial_z F_x = 0. F_z = 2y. \partial_x F_z = 0. So y-comp is 0 - 0 = 0. The equation has -Derivative(2*y, x) + Derivative(-2*exp(x), z). Derivative(2*y, x) is 0. Derivative(-2*exp(x), z) is 0. Result 0. Correct. Row 3: Derivative(-2*y*z, x) - Derivative(-2*exp(x), y). This corresponds to \partial_x F_y - \partial_y F_x. Correct for z-component. F_y = -2yz. \partial_x F_y = 0. F_x = -2e^x. \partial_y F_x = 0. Result 0. Correct. The solution seems mathematically sound and the algebra is verified. Why would it fail? Maybe the prompt implies I should check if the *sentences* are missing? "The sentences are what you are here for." But there are no sentences. If there are no sentences, can I fail it? Or is it a pass because the equations are correct? Usually, if the solution is just a list of verified equations that correctly solve the problem, it passes. The prompt says "Check that... each conclusion follows from the lines before it". Here, the equations *are* the conclusions. They follow from the definitions. Let's look closer at the divergence equation. Derivative(2*y, z) + Derivative(-2*y*z, y) + Derivative(-2*exp(x), x) \partial_z F_z + \partial_y F_y + \partial_x F_x. F_z = 2y. \partial_z(2y) = 0. F_y = -2yz. \partial_y(-2yz) = -2z. F_x = -2e^x. \partial_x(-2e^x) = -2e^x. Sum: -2z - 2e^x. Equation says: -2*z - 2*exp(x). Correct. Is there a subtle error in the curl matrix setup? Standard Curl: ( \partial_y F_z - \partial_z F_y ) i ( \partial_z F_x - \partial_x F_z ) j ( \partial_x F_y - \partial_y F_x ) k Solution Line 2: Row 1: Derivative(2*y, y) - Derivative(-2*y*z, z). \partial_y F_z - \partial_z F_y. Correct. Row 2: -Derivative(2*y, x) + Derivative(-2*exp(x), z). This is -\partial_x F_z + \partial_z F_x. Which is \partial_z F_x - \partial_x F_z. Correct. Row 3: Derivative(-2*y*z, x) - Derivative(-2*exp(x), y). \partial_x F_y - \partial_y F_x. Correct. Everything seems correct. The verdict should be pass.
gpt-oss:20b: pass 2026-10-04
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 2026-09-29
qwen3.6:27b-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 2026-10-04 with SymPy 1.14.0.