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Divergence and curl at a point

Problem 12.467 · 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 \( \displaystyle (0, 1, 2) \).
  1. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} 3 y z \right|_{\substack{ x=0\\ y=1\\ z=2 }} = 6 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} 3 y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} 3 y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\end{matrix}\right] = \left[\begin{matrix}-3\\-4\\0\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 6,\quad \nabla\times\mathbf F = \left\langle -3, -4, 0 \right\rangle \)

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 formula used in line 2 is incorrect. The standard definition is \(\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\), but the solution computes the second component as \(P_z - R_x\) (correct) and the third as \(Q_x - P_y\) (correct), yet the first component is listed as \(R_y - Q_z\) in the comment but the matrix entry shows \(\partial_y(x z^2) - \partial_z(3 y z) = 0 - 3y = -3\). Wait, let's re-evaluate. \(\mathbf{F} = \langle x^2 y, 3yz, xz^2 \rangle\). \(\nabla \times \mathbf{F} = \langle \partial_y(xz^2) - \partial_z(3yz), \partial_z(x^2y) - \partial_x(xz^2), \partial_x(3yz) - \partial_y(x^2y) \rangle\). Component 1: \(\partial_y(xz^2) - \partial_z(3yz) = 0 - 3y = -3y\). At (0,1,2), this is -3. The solution has -3. Correct. Component 2: \(\partial_z(x^2y) - \partial_x(xz^2) = 0 - z^2 = -z^2\). At (0,1,2), this is -4. The solution has -4. Correct. Component 3: \(\partial_x(3yz) - \partial_y(x^2y) = 0 - x^2 = -x^2\). At (0,1,2), this is 0. The solution has 0. Correct. The comment says: `curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩`. This is the correct formula. The matrix entries are: 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), ...)` -> \(-R_x + P_z\). Correct. Row 3: `Subs(-Derivative(x**2*y, y) + Derivative(3*y*z, x), ...)` -> \(-P_y + Q_x\). Correct. The calculations are correct. The setup is correct. The result is correct. Why did I think it was wrong? Let me double check the divergence. \(\nabla \cdot \mathbf{F} = \partial_x(x^2 y) + \partial_y(3yz) + \partial_z(xz^2) = 2xy + 3z + 2xz\). At (0,1,2): \(2(0)(1) + 3(2) + 2(0)(2) = 0 + 6 + 0 = 6\). Correct. The solution is actually correct. The comment in line 2 matches the matrix entries. The matrix entries match the standard curl formula. The evaluation is correct. The divergence is correct. Wait, look closely at Line 2's comment: `curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩`. Look at Line 2's matrix: Row 2: `-Derivative(x*z**2, x) + Derivative(x**2*y, z)`. This is \(-R_x + P_z\). This matches \(P_z - R_x\). Row 3: `-Derivative(x**2*y, y) + Derivative(3*y*z, x)`. This is \(-P_y + Q_x\). This matches \(Q_x - P_y\). Everything seems correct. Is there a subtle error? Let's check the partials again. \(P = x^2 y, Q = 3yz, R = xz^2\). \(P_x = 2xy, Q_y = 3z, R_z = 2xz\). Div = \(2xy + 3z + 2xz\). At (0,1,2) -> 6. Correct. Curl: \(x\)-comp: \(R_y - Q_z = \partial_y(xz^2) - \partial_z(3yz) = 0 - 3y = -3y\). At (0,1,2) -> -3. Correct. \(y\)-comp: \(P_z - R_x = \partial_z(x^2y) - \partial_x(xz^2) = 0 - z^2 = -z^2\). At (0,1,2) -> -4. Correct. \(z\)-comp: \(Q_x - P_y = \partial_x(3yz) - \partial_y(x^2y) = 0 - x^2 = -x^2\). At (0,1,2) -> 0. Correct. The solution is sound. I will mark it as pass.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-19 — The curl formula used in line 2 is incorrect. The standard definition is \(\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\), but the solution computes the second component as \(P_z - R_x\) (correct) and the third as \(Q_x - P_y\) (correct), yet the first component is listed as \(R_y - Q_z\) in the comment but the matrix entry shows \(\partial_y(x z^2) - \partial_z(3 y z) = 0 - 3y = -3\). Wait, let's re-evaluate. \(\mathbf{F} = \langle x^2 y, 3yz, xz^2 \rangle\). \(\nabla \times \mathbf{F} = \langle \partial_y(xz^2) - \partial_z(3yz), \partial_z(x^2y) - \partial_x(xz^2), \partial_x(3yz) - \partial_y(x^2y) \rangle\). Component 1: \(\partial_y(xz^2) - \partial_z(3yz) = 0 - 3y = -3y\). At (0,1,2), this is -3. The solution has -3. Correct. Component 2: \(\partial_z(x^2y) - \partial_x(xz^2) = 0 - z^2 = -z^2\). At (0,1,2), this is -4. The solution has -4. Correct. Component 3: \(\partial_x(3yz) - \partial_y(x^2y) = 0 - x^2 = -x^2\). At (0,1,2), this is 0. The solution has 0. Correct. The comment says: `curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩`. This is the correct formula. The matrix entries are: 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), ...)` -> \(-R_x + P_z\). Correct. Row 3: `Subs(-Derivative(x**2*y, y) + Derivative(3*y*z, x), ...)` -> \(-P_y + Q_x\). Correct. The calculations are correct. The setup is correct. The result is correct. Why did I think it was wrong? Let me double check the divergence. \(\nabla \cdot \mathbf{F} = \partial_x(x^2 y) + \partial_y(3yz) + \partial_z(xz^2) = 2xy + 3z + 2xz\). At (0,1,2): \(2(0)(1) + 3(2) + 2(0)(2) = 0 + 6 + 0 = 6\). Correct. The solution is actually correct. The comment in line 2 matches the matrix entries. The matrix entries match the standard curl formula. The evaluation is correct. The divergence is correct. Wait, look closely at Line 2's comment: `curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩`. Look at Line 2's matrix: Row 2: `-Derivative(x*z**2, x) + Derivative(x**2*y, z)`. This is \(-R_x + P_z\). This matches \(P_z - R_x\). Row 3: `-Derivative(x**2*y, y) + Derivative(3*y*z, x)`. This is \(-P_y + Q_x\). This matches \(Q_x - P_y\). Everything seems correct. Is there a subtle error? Let's check the partials again. \(P = x^2 y, Q = 3yz, R = xz^2\). \(P_x = 2xy, Q_y = 3z, R_z = 2xz\). Div = \(2xy + 3z + 2xz\). At (0,1,2) -> 6. Correct. Curl: \(x\)-comp: \(R_y - Q_z = \partial_y(xz^2) - \partial_z(3yz) = 0 - 3y = -3y\). At (0,1,2) -> -3. Correct. \(y\)-comp: \(P_z - R_x = \partial_z(x^2y) - \partial_x(xz^2) = 0 - z^2 = -z^2\). At (0,1,2) -> -4. Correct. \(z\)-comp: \(Q_x - P_y = \partial_x(3yz) - \partial_y(x^2y) = 0 - x^2 = -x^2\). At (0,1,2) -> 0. Correct. The solution is sound. I will mark it as pass.
  • 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: fail (error) 2026-10-17 — 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 y-component as -(R_x) + P_z (correct) but the z-component as -(P_y) + Q_x (correct). However, looking closely at the matrix provided: Row 1 is R_y - Q_z. Row 2 is -R_x + P_z. Row 3 is -P_y + Q_x. Wait, let's re-evaluate the values. F = <x^2y, 3yz, xz^2>. P = x^2y, Q = 3yz, R = xz^2. Divergence: P_x = 2xy, Q_y = 3z, R_z = 2xz. At (0,1,2): 0 + 6 + 0 = 6. Correct. Curl: Component 1 (x-comp): R_y - Q_z. R_y = 0. Q_z = 3y. At (0,1,2): 0 - 3(1) = -3. Correct. Component 2 (y-comp): P_z - R_x. P_z = 0. R_x = z^2. At (0,1,2): 0 - (2^2) = -4. Correct. Component 3 (z-comp): Q_x - P_y. Q_x = 0. P_y = x^2. At (0,1,2): 0 - 0^2 = 0. Correct. The solution's code snippets: Row 1: Derivative(x*z**2, y) - Derivative(3*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(3*y*z, x) -> -P_y + Q_x. Correct. The results are -3, -4, 0. The stated answer is <-3, -4, 0>. The solution is actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. "Find what is wrong". If it's correct, I should pass. Let's double check the Z-component formula in the text comment: "curl F = <R_y − Q_z, P_z − R_x, Q_x − P_y>". This is the standard correct formula. The code implements: 1. R_y - Q_z 2. P_z - R_x (written as -R_x + P_z) 3. Q_x - P_y (written as -P_y + Q_x) All steps are algebraically correct and evaluate to the correct numbers. The setup is correct. The conclusion is correct.
  • 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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