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Divergence and curl at a point

Problem 12.434 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, y z, x z^{2} \right\rangle \) at \( \displaystyle (1, 1, 1) \).
  1. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} y z \right|_{\substack{ x=1\\ y=1\\ z=1 }} = 5 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}-1\\-1\\-1\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 5,\quad \nabla\times\mathbf F = \left\langle -1, -1, -1 \right\rangle \)

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 z-component should be Q_x - P_y = 0 - x^2 = -1, but the y-component is calculated as P_z - R_x = 0 - z^2 = -1, whereas the formula used in the text (P_z - R_x) yields 0 - 1 = -1, which matches, but wait: P_z = 0, R_x = z^2. At (1,1,1), R_x = 1. So P_z - R_x = -1. The x-component is R_y - Q_z = 0 - y = -1. The z-component is Q_x - P_y = 0 - x^2 = -1. The result < -1, -1, -1 > is actually correct. Let me re-read the matrix definition in line 2. Line 2 defines the matrix entries as: Row 1: R_y - Q_z. Row 2: -R_x + P_z (which is P_z - R_x). Row 3: -P_y + Q_x (which is Q_x - P_y). This matches the standard curl formula < R_y - Q_z, P_z - R_x, Q_x - P_y >. The values are: R_y = 0, Q_z = y = 1 -> -1. P_z = 0, R_x = z^2 = 1 -> -1. Q_x = 0, P_y = x^2 = 1 -> -1. The calculation is correct. Why did I think it was wrong? Ah, I misread the vector field or the derivatives in my head. Let's double check. F = <x^2y, yz, xz^2>. div = 2xy + z + 2xz. At (1,1,1): 2+1+2=5. Correct. curl = < d/dy(xz^2)-d/dz(yz), d/dz(x^2y)-d/dx(xz^2), d/dx(yz)-d/dy(x^2y) > = < 0-y, 0-z^2, 0-x^2 > = < -y, -z^2, -x^2 >. At (1,1,1): < -1, -1, -1 >. The solution is correct. The text description in line 2 says curl F = < R_y - Q_z, P_z - R_x, Q_x - P_y >. This is the standard formula. The matrix entries match this formula. The substitution is correct. The result is correct. So why would I fail it? I shouldn't. It is correct.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-18 — The curl calculation is incorrect. The z-component should be Q_x - P_y = 0 - x^2 = -1, but the y-component is calculated as P_z - R_x = 0 - z^2 = -1, whereas the formula used in the text (P_z - R_x) yields 0 - 1 = -1, which matches, but wait: P_z = 0, R_x = z^2. At (1,1,1), R_x = 1. So P_z - R_x = -1. The x-component is R_y - Q_z = 0 - y = -1. The z-component is Q_x - P_y = 0 - x^2 = -1. The result < -1, -1, -1 > is actually correct. Let me re-read the matrix definition in line 2. Line 2 defines the matrix entries as: Row 1: R_y - Q_z. Row 2: -R_x + P_z (which is P_z - R_x). Row 3: -P_y + Q_x (which is Q_x - P_y). This matches the standard curl formula < R_y - Q_z, P_z - R_x, Q_x - P_y >. The values are: R_y = 0, Q_z = y = 1 -> -1. P_z = 0, R_x = z^2 = 1 -> -1. Q_x = 0, P_y = x^2 = 1 -> -1. The calculation is correct. Why did I think it was wrong? Ah, I misread the vector field or the derivatives in my head. Let's double check. F = <x^2y, yz, xz^2>. div = 2xy + z + 2xz. At (1,1,1): 2+1+2=5. Correct. curl = < d/dy(xz^2)-d/dz(yz), d/dz(x^2y)-d/dx(xz^2), d/dx(yz)-d/dy(x^2y) > = < 0-y, 0-z^2, 0-x^2 > = < -y, -z^2, -x^2 >. At (1,1,1): < -1, -1, -1 >. The solution is correct. The text description in line 2 says curl F = < R_y - Q_z, P_z - R_x, Q_x - P_y >. This is the standard formula. The matrix entries match this formula. The substitution is correct. The result is correct. So why would I fail it? I shouldn't. It is correct.
  • gpt-oss:20b: pass 2026-10-18
  • qwen3.6:27b-mlx: pass 2026-10-16
  • gpt-oss:20b: pass 2026-10-16
  • qwen3.6:27b-mlx: fail (error) 2026-10-15 — The curl calculation is incorrect. The z-component should be Q_x - P_y = y - 2xy, which evaluates to -1 at (1,1,1). However, the x-component is R_y - Q_z = 0 - y = -1, and the y-component is P_z - R_x = 0 - z^2 = -1. Wait, let's re-evaluate carefully. F = <x^2y, yz, xz^2> P = x^2y, Q = yz, R = xz^2 Div: P_x + Q_y + R_z = 2xy + z + 2xz. At (1,1,1): 2(1)(1) + 1 + 2(1)(1) = 2 + 1 + 2 = 5. This is correct. Curl: x-comp: R_y - Q_z = (xz^2)_y - (yz)_z = 0 - y = -y. At (1,1,1) -> -1. y-comp: P_z - R_x = (x^2y)_z - (xz^2)_x = 0 - z^2 = -z^2. At (1,1,1) -> -1. z-comp: Q_x - P_y = (yz)_x - (x^2y)_y = 0 - x^2 = -x^2. At (1,1,1) -> -1. So the curl is <-1, -1, -1>. The stated answer is correct. The solution code calculates: Row 1: Derivative(x*z**2, y) - Derivative(y*z, z) = 0 - y = -y. Correct. Row 2: -Derivative(x*z**2, x) + Derivative(x**2*y, z) = -z^2 + 0 = -z^2. Correct. Row 3: -Derivative(x**2*y, y) + Derivative(y*z, x) = -x^2 + 0 = -x^2. Correct. Wait, the standard formula for curl is <R_y - Q_z, P_z - R_x, Q_x - P_y>. The code uses: Row 1: R_y - Q_z. Correct. Row 2: -R_x + P_z. Correct. Row 3: -P_y + Q_x. Correct. The evaluation at (1,1,1) gives <-1, -1, -1>. The stated answer is correct. The solution steps are correct. Why did I think it was wrong? I misread the z-component in my head initially. Let me double check the problem statement vs the solution. Problem: F = <x^2 y, y z, x z^2> Div = 2xy + z + 2xz. At (1,1,1) = 5. Correct. Curl x = d/dy(xz^2) - d/dz(yz) = 0 - y = -y = -1. Curl y = d/dz(x^2y) - d/dx(xz^2) = 0 - z^2 = -1. Curl z = d/dx(yz) - d/dy(x^2y) = 0 - x^2 = -1. The solution is actually correct. The verdict should be pass.
  • gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.

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