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Divergence and curl at a point

Problem 12.471 · 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 \( \displaystyle (1, 2, -1) \).
  1. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }} = 5 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} \left(- 3 y z\right) \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\end{matrix}\right] = \left[\begin{matrix}6\\-1\\-1\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 5,\quad \nabla\times\mathbf F = \left\langle 6, -1, -1 \right\rangle \)

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 in line 2 uses the wrong formula. The components are calculated as (R_y - Q_z), (P_z - R_x), and (Q_x - P_y), but the standard curl formula is (R_y - Q_z), (P_z - R_x) is incorrect for the y-component (should be R_x - P_z or -(P_z - R_x) depending on convention, but typically curl_y = R_x - P_z), and the z-component is Q_x - P_y. Specifically, the y-component of the curl is R_x - P_z. The solution computes P_z - R_x, which is the negative of the correct value. Let's check: F = <x^2y, -3yz, xz^2>. curl_x = d/dy(xz^2) - d/dz(-3yz) = 0 - (-3y) = 3y. At (1,2,-1), 3(2)=6. Correct. curl_y = d/dz(x^2y) - d/dx(xz^2) = 0 - z^2 = -z^2. At (1,2,-1), -(-1)^2 = -1. The solution computes -d/dx(xz^2) + d/dz(x^2y) = -z^2 + 0 = -1. This matches. curl_z = d/dx(-3yz) - d/dy(x^2y) = 0 - x^2 = -x^2. At (1,2,-1), -1^2 = -1. The solution computes -d/dy(x^2y) + d/dx(-3yz) = -x^2 + 0 = -1. This matches. Wait, let me re-read the standard formula. curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y> is WRONG. The standard determinant form is i(R_y - Q_z) - j(R_x - P_z) + k(Q_x - P_y). So the y-component is -(R_x - P_z) = P_z - R_x. The solution uses P_z - R_x. So the formula used in the comment is actually correct for the vector components if written as <..., P_z - R_x, ...>. Let's re-verify the values. curl_x = 3y = 6. curl_y = P_z - R_x = 0 - z^2 = -1. curl_z = Q_x - P_y = 0 - x^2 = -1. The result <6, -1, -1> is correct. The comment says curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. This is the correct formula for the components. The calculation is correct. Why did I think it was wrong? I confused the sign of the middle term in the determinant expansion. The determinant is |i j k; d/dx d/dy d/dz; P Q R|. i(R_y - Q_z) - j(P_z - R_x) + k(Q_x - P_y) = <R_y - Q_z, R_x - P_z, Q_x - P_y>. Ah! The standard vector component for y is R_x - P_z. The solution uses P_z - R_x. P_z - R_x = -(R_x - P_z). So the solution has the wrong sign for the y-component in the formula description, but let's check the calculation. The calculation line is: Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...). This is -R_x + P_z = P_z - R_x. The value is -1. The correct value for curl_y is R_x - P_z = z^2 - 0 = 1. So the correct curl_y is 1. The solution gives -1. The formula in the comment is <..., P_z - R_x, ...> which yields -1. The standard formula is <..., R_x - P_z, ...> which yields 1. The solution is incorrect.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-19 — The curl calculation in line 2 uses the wrong formula. The components are calculated as (R_y - Q_z), (P_z - R_x), and (Q_x - P_y), but the standard curl formula is (R_y - Q_z), (P_z - R_x) is incorrect for the y-component (should be R_x - P_z or -(P_z - R_x) depending on convention, but typically curl_y = R_x - P_z), and the z-component is Q_x - P_y. Specifically, the y-component of the curl is R_x - P_z. The solution computes P_z - R_x, which is the negative of the correct value. Let's check: F = <x^2y, -3yz, xz^2>. curl_x = d/dy(xz^2) - d/dz(-3yz) = 0 - (-3y) = 3y. At (1,2,-1), 3(2)=6. Correct. curl_y = d/dz(x^2y) - d/dx(xz^2) = 0 - z^2 = -z^2. At (1,2,-1), -(-1)^2 = -1. The solution computes -d/dx(xz^2) + d/dz(x^2y) = -z^2 + 0 = -1. This matches. curl_z = d/dx(-3yz) - d/dy(x^2y) = 0 - x^2 = -x^2. At (1,2,-1), -1^2 = -1. The solution computes -d/dy(x^2y) + d/dx(-3yz) = -x^2 + 0 = -1. This matches. Wait, let me re-read the standard formula. curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y> is WRONG. The standard determinant form is i(R_y - Q_z) - j(R_x - P_z) + k(Q_x - P_y). So the y-component is -(R_x - P_z) = P_z - R_x. The solution uses P_z - R_x. So the formula used in the comment is actually correct for the vector components if written as <..., P_z - R_x, ...>. Let's re-verify the values. curl_x = 3y = 6. curl_y = P_z - R_x = 0 - z^2 = -1. curl_z = Q_x - P_y = 0 - x^2 = -1. The result <6, -1, -1> is correct. The comment says curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. This is the correct formula for the components. The calculation is correct. Why did I think it was wrong? I confused the sign of the middle term in the determinant expansion. The determinant is |i j k; d/dx d/dy d/dz; P Q R|. i(R_y - Q_z) - j(P_z - R_x) + k(Q_x - P_y) = <R_y - Q_z, R_x - P_z, Q_x - P_y>. Ah! The standard vector component for y is R_x - P_z. The solution uses P_z - R_x. P_z - R_x = -(R_x - P_z). So the solution has the wrong sign for the y-component in the formula description, but let's check the calculation. The calculation line is: Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...). This is -R_x + P_z = P_z - R_x. The value is -1. The correct value for curl_y is R_x - P_z = z^2 - 0 = 1. So the correct curl_y is 1. The solution gives -1. The formula in the comment is <..., P_z - R_x, ...> which yields -1. The standard formula is <..., R_x - P_z, ...> which yields 1. The solution is incorrect.
  • 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 specified point are correct.
  • gpt-oss:20b: pass 2026-10-17
  • qwen3.6:27b-mlx: inconclusive 2026-10-17 — 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 2026-10-17

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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