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Divergence and curl at a point

Problem 12.466 · 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. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=1\\ z=1 }} = 1 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=1\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}3\\-1\\-1\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 1,\quad \nabla\times\mathbf F = \left\langle 3, -1, -1 \right\rangle \)

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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The curl calculation in line 2 uses an incorrect formula. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution compute
Every verdict on record (6)
  • qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The curl calculation in line 2 uses an incorrect formula. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution compute
  • gpt-oss:20b: pass 2026-10-19
  • qwen3.6:27b-mlx: pass 2026-10-17 — 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 2026-10-17
  • qwen3.6:27b-mlx: fail (error) 2026-10-17 — The curl calculation in line 2 uses an incorrect formula. The second component should be R_x - P_z, but the solution computes P_z - R_x (which is the negative of the correct component). The third component is also incorrect: it computes Q_x - P_y instead of P_y - Q_x. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y> is WRONG; it should be <R_y - Q_z, P_z - R_x, Q_x - P_y> wait, let me re-verify standard definition. Standard curl: i( R_y - Q_z ) - j( R_x - P_z ) + k( Q_x - P_y ) = < R_y - Q_z, P_z - R_x, Q_x - P_y >. Let's check the solution's matrix: Row 1: R_y - Q_z. Correct. Row 2: -R_x + P_z = P_z - R_x. Correct. Row 3: -P_y + Q_x = Q_x - P_y. Correct. Wait, let me re-read the solution's matrix carefully. Solution Row 2: `Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...` `-Derivative(x*z**2, x)` is `-R_x`. `Derivative(x**2*y, z)` is `P_z`. So Row 2 is `P_z - R_x`. This is correct. Solution Row 3: `Subs(-Derivative(x**2*y, y) + Derivative(-3*y*z, x)...` `-Derivative(x**2*y, y)` is `-P_y`. `Derivative(-3*y*z, x)` is `Q_x`. So Row 3 is `Q_x - P_y`. This is correct. Let's re-calculate the values to be sure. F = < x^2 y, -3 y z, x z^2 > P = x^2 y, Q = -3 y z, R = x z^2 Divergence: P_x = 2xy Q_y = -3z R_z = 2xz Div = 2xy - 3z + 2xz At (1,1,1): 2(1)(1) - 3(1) + 2(1)(1) = 2 - 3 + 2 = 1. Correct. Curl: Component 1 (i): R_y - Q_z R_y = d/dy(x z^2) = 0 Q_z = d/dz(-3 y z) = -3y R_y - Q_z = 0 - (-3y) = 3y At (1,1,1): 3(1) = 3. Correct. Component 2 (j): P_z - R_x P_z = d/dz(x^2 y) = 0 R_x = d/dx(x z^2) = z^2 P_z - R_x = 0 - z^2 = -z^2 At (1,1,1): -1. Correct. Component 3 (k): Q_x - P_y Q_x = d/dx(-3 y z) = 0 P_y = d/dy(x^2 y) = x^2 Q_x - P_y = 0 - x^2 = -x^2 At (1,1,1): -1. Correct. The solution's code and result are actually correct. My initial quick scan was wrong. The formula used in the comment `curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩` is the standard correct formula. The implementation in the matrix matches this formula. The numerical results are correct. Therefore, the solution is sound.
  • gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.

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