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Divergence and curl at a point

Problem 12.352 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, 3 y z, x z^{2} \right\rangle \) at \( \displaystyle (1, 1, 1) \).
  1. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} 3 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }} = 7 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} 3 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} 3 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}-3\\-1\\-1\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 7,\quad \nabla\times\mathbf F = \left\langle -3, -1, -1 \right\rangle \)

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
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
2✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0central differences of F at the point

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: fail (error) — The curl calculation is incorrect. The second component should be $P_z - R_x = 2xz - z^2$, which evaluates to $1$ at $(1,1,1)$, but the solution computes $-1$. The third component should be $Q_x - P_y = 0 - x^2$, which evaluates to $-1$, but the solution computes $-1$ via an incorrect formula ($-P_y + Q_x$ is correct, but the terms in the matrix line are swapped or mislabeled relative to standard definitions, leading to the wrong sign for the second component specifically).
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-17 — The curl calculation is incorrect. The second component should be $P_z - R_x = 2xz - z^2$, which evaluates to $1$ at $(1,1,1)$, but the solution computes $-1$. The third component should be $Q_x - P_y = 0 - x^2$, which evaluates to $-1$, but the solution computes $-1$ via an incorrect formula ($-P_y + Q_x$ is correct, but the terms in the matrix line are swapped or mislabeled relative to standard definitions, leading to the wrong sign for the second component specifically).
  • gpt-oss:20b: pass 2026-10-17
  • qwen3.6:27b-mlx: fail (error) 2026-10-15 — The curl calculation is incorrect. The first component should be R_y - Q_z = 0 - 3x = -3, but the second component is P_z - R_x = 0 - z^2 = -1, and the third is Q_x - P_y = 0 - x^2 = -1. Wait, let's re-calculate carefully. F = <x^2y, 3yz, xz^2>. div F = 2xy + 3z + 2xz. At (1,1,1): 2+3+2=7. Correct. curl F = <d/dy(xz^2) - d/dz(3yz), d/dz(x^2y) - d/dx(xz^2), d/dx(3yz) - d/dy(x^2y)>. Component 1: 0 - 3x = -3x. At (1,1,1) -> -3. Component 2: 0 - z^2 = -z^2. At (1,1,1) -> -1. Component 3: 0 - x^2 = -x^2. At (1,1,1) -> -1. The stated answer is <-3, -1, -1>. The solution code calculates: Row 1: d/dy(xz^2) - d/dz(3yz) = 0 - 3x = -3. Row 2: -d/dx(xz^2) + d/dz(x^2y) = -z^2 + 0 = -1. Row 3: -d/dy(x^2y) + d/dx(3yz) = -x^2 + 0 = -1. The code and result are actually correct. Why did I think it was wrong? Let me re-read the prompt's stated answer. Stated answer: curl F = <-3, -1, -1>. My manual calculation: <-3, -1, -1>. The solution is correct. Wait, looking at the code for Row 2: `Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...)`. Derivative(x*z**2, x) is z**2. Derivative(x**2*y, z) is 0. So -z**2 + 0 = -1. Correct. Row 3: `Subs(-Derivative(x**2*y, y) + Derivative(3*y*z, x)...)`. Derivative(x**2*y, y) is x**2. Derivative(3*y*z, x) is 0. So -x**2 + 0 = -1. Correct. The solution is correct. I will mark it as pass.
  • gpt-oss:20b: pass 2026-10-15
  • qwen3.6:27b-mlx: fail (error) 2026-10-14 — The curl calculation contains sign errors in the component definitions. The standard formula is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution computes the y-component as -(R_x) + P_z (correct) but labels it with a negative sign in the matrix construction logic or simply miscalculates the terms: P_z = 2xz = 2, R_x = z^2 = 1, so P_z - R_x = 1, not -1. Similarly for the z-component: Q_x = 0, P_y = x^2 = 1, so Q_x - P_y = -1. Wait, let's re-evaluate. F = <x^2y, 3yz, xz^2>. P = x^2y, Q = 3yz, R = xz^2. curl_x = R_y - Q_z = 0 - 3y = -3y. At (1,1,1), this is -3. Correct. curl_y = P_z - R_x = 0 - z^2 = -z^2. At (1,1,1), this is -1. Correct. curl_z = Q_x - P_y = 0 - x^2 = -x^2. At (1,1,1), this is -1. Correct. The stated answer is <-3, -1, -1>. The solution's matrix output is <-3, -1, -1>. The equations are marked 'unchecked' but the text says 'computer algebra system has already checked'. If the CAS checked the algebra, the result is correct. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The lines provided are labeled [equation, unchecked]. This is contradictory. Usually, 'unchecked' means I should check it. Let's look closely at the matrix definition in line 2. Line 2 defines the matrix entries as: Row 1: Subs(Derivative(x*z**2, y) - Derivative(3*y*z, z)...) -> R_y - Q_z. Correct. Row 2: Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...) -> -R_x + P_z. This is P_z - R_x. Correct. Row 3: Subs(-Derivative(x**2*y, y) + Derivative(3*y*z, x)...) -> -P_y + Q_x. This is Q_x - P_y. Correct. The calculations: R_y = 0. Q_z = 3y. 0 - 3(1) = -3. R_x = z^2. P_z = 0. -1^2 + 0 = -1. P_y = x^2. Q_x = 0. -1^2 + 0 = -1. The result is indeed <-3, -1, -1>. The setup is correct. The algebra is correct. The verdict should be pass.
  • gpt-oss:20b: pass 2026-10-14

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/div_curl_point, checked 2026-10-17 with SymPy 1.14.0.

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