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Divergence and curl at a point

Problem 12.387 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x^{2} y, - y z, x z^{2} \right\rangle \) at \( \displaystyle (2, -1, 3) \).
  1. \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} - y z \right|_{\substack{ x=2\\ y=-1\\ z=3 }} = 5 \]
    div F = P_x + Q_y + R_z, at the point.✓ Proved
  2. \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} - y z \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} - y z \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\end{matrix}\right] = \left[\begin{matrix}-1\\-9\\-4\end{matrix}\right] \]
    curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 5,\quad \nabla\times\mathbf F = \left\langle -1, -9, -4 \right\rangle \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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: pass
Every verdict on record (6)
  • qwen3.6:27b-mlx: pass 2026-10-18
  • gpt-oss:20b: pass 2026-10-18
  • qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.
  • gpt-oss:20b: pass 2026-10-15
  • qwen3.6:27b-mlx: pass 2026-10-14
  • gpt-oss:20b: pass 2026-10-14

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.

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