∫Calc Practice
Home›Calculus 3›Divergence and curl at a point›Problem 12.472

Divergence and curl at a point

Problem 12.472 · 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. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} 2 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }} = 6 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} 2 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} 2 y z \right|_{\substack{ x=1\\ y=1\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}-2\\-1\\-1\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 6,\quad \nabla\times\mathbf F = \left\langle -2, -1, -1 \right\rangle \)

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: fail (error) — The curl calculation is incorrect. The second component should be P_z - R_x = z^2 - z^2 = 0, not -1. The third component should be Q_x - P_y = 0 - x^2 = -1, which is correct, but the first component R_y - Q_z = 0 - 2y = -2 is also correct. The error lies in the second component of the curl vector.
Every verdict on record (6)
  • qwen3.6:27b-mlx: fail (error) 2026-10-19 — The curl calculation is incorrect. The second component should be P_z - R_x = z^2 - z^2 = 0, not -1. The third component should be Q_x - P_y = 0 - x^2 = -1, which is correct, but the first component R_y - Q_z = 0 - 2y = -2 is also correct. The error lies in the second component of the curl vector.
  • gpt-oss:20b: pass 2026-10-19
  • qwen3.6:27b-mlx: fail (error) 2026-10-17 — The curl calculation is incorrect. The first component should be R_y - Q_z = 0 - 2x = -2, but the second component is P_z - R_x = 0 - z^2 = -1, and the third is Q_x - P_y = 0 - x^2 = -1. Wait, let's re-evaluate. F = <x^2y, 2yz, xz^2>. Div: d/dx(x^2y) + d/dy(2yz) + d/dz(xz^2) = 2xy + 2z + 2xz. At (1,1,1): 2+2+2=6. Correct. Curl: (R_y - Q_z) i = (0 - 2x) i = -2x i. At (1,1,1) -> -2. (P_z - R_x) j = (0 - z^2) j = -z^2 j. At (1,1,1) -> -1. (Q_x - P_y) k = (0 - x^2) k = -x^2 k. At (1,1,1) -> -1. The stated answer is <-2, -1, -1>. The solution code calculates exactly this. Why did I think it was wrong? Let me re-read the code. Code line 2: Row 1: Subs(Derivative(x*z**2, y) - Derivative(2*y*z, z)...) -> d/dy(xz^2) - d/dz(2yz) = 0 - 2x = -2x. At (1,1,1) is -2. Correct. Row 2: Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z)...) -> -d/dx(xz^2) + d/dz(x^2y) = -z^2 + 0 = -z^2. At (1,1,1) is -1. Correct. Row 3: Subs(-Derivative(x**2*y, y) + Derivative(2*y*z, x)...) -> -d/dy(x^2y) + d/dx(2yz) = -x^2 + 0 = -x^2. At (1,1,1) is -1. Correct. The solution is actually correct. I will mark it as pass.
  • 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 P_z - R_x (correct) but labels it with a negative sign in the matrix setup or computes it incorrectly, and the z-component as Q_x - P_y (correct) but labels it with a negative sign. Specifically, the matrix shows -Derivative(x*z**2, x) + Derivative(x**2*y, z) for the second component, which is -R_x + P_z, matching the formula, but the result -1 is correct for P_z - R_x = 1 - 2 = -1. Wait, let's re-evaluate. F = <x^2y, 2yz, xz^2>. P = x^2y, Q = 2yz, R = xz^2. Curl components: 1. R_y - Q_z = 0 - 2y = -2y. At (1,1,1), this is -2. The solution has -2. Correct. 2. P_z - R_x = 0 - z^2 = -z^2. At (1,1,1), this is -1. The solution has -1. Correct. 3. Q_x - P_y = 0 - x^2 = -x^2. At (1,1,1), this is -1. The solution has -1. Correct. Let's look at the matrix code in line 2: 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 in the code is actually correct for the standard curl definition <R_y-Q_z, P_z-R_x, Q_x-P_y>. The results are correct. The text description says "curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>". This is the standard definition. Wait, I need to check if the problem asks for something else or if there's a subtle error. Divergence: P_x + Q_y + R_z = 2xy + 2z + 2xz. At (1,1,1): 2+2+2=6. Correct. Curl: < -2y, -z^2, -x^2 >. At (1,1,1): <-2, -1, -1>. Correct. The solution seems correct. Why did I think it was wrong? Let me re-read the prompt's instruction. "Your job is to find what is wrong...". If it's correct, I should pass. Let's double check the matrix indices vs standard vector components. Standard Curl: ( R_y - Q_z ) i ( P_z - R_x ) j ( Q_x - P_y ) k Solution Matrix Row 1: R_y - Q_z. Correct. Solution Matrix Row 2: -R_x + P_z. Correct. Solution Matrix Row 3: -P_y + Q_x. Correct. The solution is correct. I will mark it as pass.
  • 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.

← Problem 12.471All divergence and curl at a point problemsProblem 12.473 →