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Home›Calculus 3›Divergence and curl at a point›Problem 12.473

Divergence and curl at a point

Problem 12.473 · 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. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} \left(- 3 y z\right) \right|_{\substack{ x=0\\ y=1\\ z=2 }} = -6 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} \left(- 3 y z\right) \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} \left(- 3 y z\right) \right|_{\substack{ x=0\\ y=1\\ z=2 }}\end{matrix}\right] = \left[\begin{matrix}3\\-4\\0\end{matrix}\right] \]
    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 curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution computes the z-component as Q_x - P_y, but the code calculates -P_y + Q_x (which is correct), however, the y-component is calculated as -R_x + P_z (correct) but the x-component is R_y - Q_z (correct). Wait, let's look closer at the code vs the text. Text says: curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. Code for x-comp: Derivative(R, y) - Derivative(Q, z) -> R_y - Q_z. Correct. Code for y-comp: -Derivative(R, x) + Derivative(P, z) -> P_z - R_x. Correct. Code for z-comp: -Derivative(P, y) + Derivative(Q, x) -> Q_x - P_y. Correct. Let's re-evaluate the values. P = x^2 y, Q = -3yz, R = xz^2. Divergence: P_x = 2xy. At (0,1,2) -> 0. Q_y = -3z. At (0,1,2) -> -6. R_z = 2xz. At (0,1,2) -> 0. Div = 0 - 6 + 0 = -6. Correct. Curl: x-comp: R_y - Q_z. R_y = 0. Q_z = -3y. At (0,1,2) -> -3(1) = -3. R_y - Q_z = 0 - (-3) = 3. Correct. y-comp: P_z - R_x. P_z = 0. R_x = z^2. At (0,1,2) -> 4. P_z - R_x = 0 - 4 = -4. Correct. z-comp: Q_x - P_y. Q_x = 0. P_y = x^2. At (0,1,2) -> 0. Q_x - P_y = 0 - 0 = 0. Correct. The result <3, -4, 0> is correct. The divergence -6 is correct. Why did I think it was wrong? Let me re-read the prompt's provided solution carefully. Line 2 code: [ Subs(Derivative(x*z**2, y) - Derivative(-3*y*z, z), ... ) ] -> R_y - Q_z. Correct. [Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z), ... )] -> -R_x + P_z. Correct. [Subs(-Derivative(x**2*y, y) + Derivative(-3*y*z, x), ... )] -> -P_y + Q_x. Correct. The text annotation says: curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. This is the standard formula. Is there a sign error in the standard formula? curl F = | i j k | | Dx Dy Dz | | P Q R | = i(Dy R - Dz Q) - j(Dx R - Dz P) + k(Dx Q - Dy P) = < R_y - Q_z, P_z - R_x, Q_x - P_y >. The solution is actually correct. Wait, look at line 1 comment: "div F = P_x + Q_y + R_z". Look at line 2 comment: "curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>". The calculations are correct. The annotations are correct. Is there a subtle error? Problem: F = <x^2 y, -3yz, xz^2>. Point: (0, 1, 2). Div: 2xy - 3z + 2xz. At (0,1,2): 0 - 6 + 0 = -6. Correct. Curl x: d/dy(xz^2) - d/dz(-3yz) = 0 - (-3y) = 3y. At (0,1,2): 3(1) = 3. Correct. Curl y: d/dz(x^2y) - d/dx(xz^2) = 0 - z^2 = -z^2. At (0,1,2): -4. Correct. Curl z: d/dx(-3yz) - d/dy(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 curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solution computes the z-component as Q_x - P_y, but the code calculates -P_y + Q_x (which is correct), however, the y-component is calculated as -R_x + P_z (correct) but the x-component is R_y - Q_z (correct). Wait, let's look closer at the code vs the text. Text says: curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. Code for x-comp: Derivative(R, y) - Derivative(Q, z) -> R_y - Q_z. Correct. Code for y-comp: -Derivative(R, x) + Derivative(P, z) -> P_z - R_x. Correct. Code for z-comp: -Derivative(P, y) + Derivative(Q, x) -> Q_x - P_y. Correct. Let's re-evaluate the values. P = x^2 y, Q = -3yz, R = xz^2. Divergence: P_x = 2xy. At (0,1,2) -> 0. Q_y = -3z. At (0,1,2) -> -6. R_z = 2xz. At (0,1,2) -> 0. Div = 0 - 6 + 0 = -6. Correct. Curl: x-comp: R_y - Q_z. R_y = 0. Q_z = -3y. At (0,1,2) -> -3(1) = -3. R_y - Q_z = 0 - (-3) = 3. Correct. y-comp: P_z - R_x. P_z = 0. R_x = z^2. At (0,1,2) -> 4. P_z - R_x = 0 - 4 = -4. Correct. z-comp: Q_x - P_y. Q_x = 0. P_y = x^2. At (0,1,2) -> 0. Q_x - P_y = 0 - 0 = 0. Correct. The result <3, -4, 0> is correct. The divergence -6 is correct. Why did I think it was wrong? Let me re-read the prompt's provided solution carefully. Line 2 code: [ Subs(Derivative(x*z**2, y) - Derivative(-3*y*z, z), ... ) ] -> R_y - Q_z. Correct. [Subs(-Derivative(x*z**2, x) + Derivative(x**2*y, z), ... )] -> -R_x + P_z. Correct. [Subs(-Derivative(x**2*y, y) + Derivative(-3*y*z, x), ... )] -> -P_y + Q_x. Correct. The text annotation says: curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. This is the standard formula. Is there a sign error in the standard formula? curl F = | i j k | | Dx Dy Dz | | P Q R | = i(Dy R - Dz Q) - j(Dx R - Dz P) + k(Dx Q - Dy P) = < R_y - Q_z, P_z - R_x, Q_x - P_y >. The solution is actually correct. Wait, look at line 1 comment: "div F = P_x + Q_y + R_z". Look at line 2 comment: "curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>". The calculations are correct. The annotations are correct. Is there a subtle error? Problem: F = <x^2 y, -3yz, xz^2>. Point: (0, 1, 2). Div: 2xy - 3z + 2xz. At (0,1,2): 0 - 6 + 0 = -6. Correct. Curl x: d/dy(xz^2) - d/dz(-3yz) = 0 - (-3y) = 3y. At (0,1,2): 3(1) = 3. Correct. Curl y: d/dz(x^2y) - d/dx(xz^2) = 0 - z^2 = -z^2. At (0,1,2): -4. Correct. Curl z: d/dx(-3yz) - d/dy(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: fail (error) 2026-10-17 — The curl calculation in line 2 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 z-component as Q_x - P_y but writes the code as -P_y + Q_x (which is correct algebraically) but the y-component code is -R_x + P_z (correct) and x-component is R_y - Q_z (correct). Wait, let's re-read the code carefully. Line 2 code: Row 1 (x-comp): Derivative(x*z**2, y) - Derivative(-3*y*z, z). R = x*z^2, Q = -3yz. R_y = 0. Q_z = -3y. R_y - Q_z = 0 - (-3y) = 3y. At (0,1,2), 3(1) = 3. Correct. Row 2 (y-comp): -Derivative(x*z**2, x) + Derivative(x**2*y, z). Standard formula: P_z - R_x. P = x^2y, R = x*z^2. P_z = 0. R_x = z^2. P_z - R_x = -z^2. At (0,1,2), -4. Code: -R_x + P_z. This matches P_z - R_x. Correct. Row 3 (z-comp): -Derivative(x**2*y, y) + Derivative(-3*y*z, x). Standard formula: Q_x - P_y. Q = -3yz, P = x^2y. Q_x = 0. P_y = x^2. Q_x - P_y = -x^2. At (0,1,2), 0. Code: -P_y + Q_x. This matches Q_x - P_y. Correct. Wait, looking at the comment in line 2: `curl F = <R_y − Q_z, P_z − R_x, Q_x − P_y>`. This is the correct formula. Let's check the algebraic evaluation again. Div: P_x + Q_y + R_z = 2xy - 3z + 2xz. At (0,1,2): 0 - 6 + 0 = -6. Correct. Curl: x-comp: R_y - Q_z = 0 - (-3y) = 3y. At (0,1,2) -> 3. Correct. y-comp: P_z - R_x = 0 - z^2 = -z^2. At (0,1,2) -> -4. Correct. z-comp: Q_x - P_y = 0 - x^2 = -x^2. At (0,1,2) -> 0. Correct. 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 divergence formula in line 1 comment: `div F = P_x + Q_y + R_z`. Correct. Let's double check the curl formula in line 2 comment: `curl F = <R_y − Q_z, P_z − R_x, Q_x − P_y>`. Correct. The equations are marked [proved]. The sentences are just comments explaining the equations. The comments are correct. The setup is correct. The result is correct. Verdict: pass.
  • gpt-oss:20b: pass 2026-10-17
  • qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly computes the divergence and curl components and evaluates them at the specified point. The algebraic steps, though presented as unchecked equations, are correct and lead to the stated answer.
  • 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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