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Divergence and curl at a point

Problem 12.489 · easy

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

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: pass — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the point (1, 1, 1) are correct.
Every verdict on record (6)
  • qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the point (1, 1, 1) are correct.
  • gpt-oss:20b: pass 2026-10-21
  • qwen3.6:27b-mlx: fail (error) 2026-10-20 — The curl calculation in line 2 is incorrect. The first component is calculated as R_y - Q_z, but the correct formula is Q_z - R_y. The second component uses P_z - R_x (correct), but the third component uses Q_x - P_y (correct). However, the first component error leads to the wrong vector <2, -1, -1> instead of the correct <-2, -1, -1>. Specifically, R_y = 0 and Q_z = -2, so Q_z - R_y = -2, but the solution computes 0 - (-2) = 2.
  • gpt-oss:20b: pass 2026-10-20
  • qwen3.6:27b-mlx: fail (error) 2026-10-20 — The curl calculation contains sign errors in the component formulas. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution computes the second component as -R_x + P_z (correct) but the third component as -P_y + Q_x (correct). Wait, let's re-read carefully. Standard Curl: 1. R_y - Q_z 2. P_z - R_x 3. Q_x - P_y Solution Line 2: Row 1: Derivative(x*z**2, y) - Derivative(-2*y*z, z) -> R_y - Q_z. Correct. Row 2: -Derivative(x*z**2, x) + Derivative(x**2*y, z) -> -R_x + P_z. Correct. Row 3: -Derivative(x**2*y, y) + Derivative(-2*y*z, x) -> -P_y + Q_x. Correct. Let's check the values: F = <x^2y, -2yz, xz^2> P = x^2y, Q = -2yz, R = xz^2 Divergence: P_x = 2xy Q_y = -2z R_z = 2xz Div = 2xy - 2z + 2xz At (1,1,1): 2(1)(1) - 2(1) + 2(1)(1) = 2 - 2 + 2 = 2. Correct. Curl: 1. R_y - Q_z = 0 - (-2y) = 2y. At (1,1,1) -> 2. 2. P_z - R_x = 0 - z^2 = -z^2. At (1,1,1) -> -1. 3. Q_x - P_y = 0 - x^2 = -x^2. At (1,1,1) -> -1. Result: <2, -1, -1>. Correct. Wait, why did I think it was wrong? Let me re-read the solution text. "curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩" This is the correct formula. Let's look at the matrix entries again. Row 1: Subs(Derivative(x*z**2, y) - Derivative(-2*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(-2*y*z, x)...) -> -P_y + Q_x. This is Q_x - P_y. Correct. The algebraic setup is correct. The evaluation is correct. The stated answer is correct. Is there a subtle error? "Subs((Derivative(x**2*y, x) + Derivative(-2*y*z, y)) + Derivative(x*z**2, z)..." This calculates P_x + Q_y + R_z. Correct. The solution seems perfectly correct. Why would it fail? Maybe the prompt implies I should find an error if one exists, but if none exists, I pass. Let's double check the curl formula signs again. curl F = | i j k | | dx dy dz | | P Q R | = i (dR/dy - dQ/dz) - j (dR/dx - dQ/dz?? No, dR/dx - dP/dz) + k (dQ/dx - dP/dy) = < R_y - Q_z, -(R_x - P_z), Q_x - P_y > = < R_y - Q_z, P_z - R_x, Q_x - P_y > Solution uses: 1. R_y - Q_z 2. -R_x + P_z (which is P_z - R_x) 3. -P_y + Q_x (which is Q_x - P_y) Everything is correct. Verdict: pass.
  • gpt-oss:20b: pass 2026-10-20

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 2026-10-21 with SymPy 1.14.0.

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