Home›Calculus 3›Divergence and curl at a point›Problem 12.432

Divergence and curl at a point

Problem 12.432 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, - y z, x z^{2} \right\rangle \) at \( \displaystyle (1, 2, -1) \).

\[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} - y z \right|_{\substack{ x=1\\ y=2\\ z=-1 }} = 3 \]
div F = P_x + Q_y + R_z, at the point.✓ Proved
\[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} - y z \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} - y z \right|_{\substack{ x=1\\ y=2\\ z=-1 }}\end{matrix}\right] = \left[\begin{matrix}2\\-1\\-1\end{matrix}\right] \]
curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved

Answer \( \nabla\cdot\mathbf F = 3,\quad \nabla\times\mathbf F = \left\langle 2, -1, -1 \right\rangle \)

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

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	central differences of F at the point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the divergence and curl of the vector field at the specified point. The algebraic steps are verified, and the final results match the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly computes the divergence and curl of the vector field at the specified point. The algebraic steps are verified, and the final results match the stated answer.
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 of the curl is Q_x - P_y = 0 - x^2 = -1 at (1,2,-1), but the y-component is P_z - R_x = 0 - z^2 = -1, not -1 as listed? Wait, let's recheck. P=x^2y, Q=-yz, R=xz^2. Curl = <R_y - Q_z, P_z - R_x, Q_x - P_y>. R_y = 0, Q_z = -y => R_y - Q_z = y = 2. P_z = 0, R_x = z^2 => P_z - R_x = -z^2 = -(-1)^2 = -1. Q_x = 0, P_y = x^2 => Q_x - P_y = -x^2 = -1. So curl is <2, -1, -1>. The stated answer is correct. The solution lines are just algebraic substitutions. Why did I think it was wrong? Let me re-read the prompt. "Find what is wrong". The solution lines are marked [equation, unchecked] but the prompt says "a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)". Wait, the lines are marked "unchecked" in the text provided by the user, but the instructions say "Each line... is either an EQUATION... or a SENTENCE... do not re-check the algebra of a line marked proved". The user's input format is slightly ambiguous. It says "1. [equation, unchecked] ...". Usually, in these tasks, if it's an equation line, I shouldn't check the algebra. However, looking closely at the provided solution text: it contains NO sentences. It only contains equations. The instructions say "The sentences are what you are here for." If there are no sentences, is it a pass? Or is the lack of explanation a style issue? Or is the fact that the curl formula in the comment is wrong? Let's check the comment in line 2: "curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩". This is the correct formula. The calculation in the matrix: 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). The algebra checks out. The result is correct. The setup is correct. There are no sentences to critique. Is it a pass? Yes. The solution is just the calculation steps. They are correct.
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.