Divergence and curl at a point
Problem 12.341 · easy
Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle \sin{\left(y \right)}, \cos{\left(x \right)}, x y z \right\rangle \) at \( \displaystyle (1, 1, 1) \).
- \[ \left. \frac{\partial}{\partial z} x y z + \frac{d}{d x} \sin{\left(y \right)} + \frac{d}{d y} \cos{\left(x \right)} \right|_{\substack{ x=1\\ y=1\\ z=1 }} = 1 \]div F = P_x + Q_y + R_z, at the point.✓ Proved
- \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x y z - \frac{d}{d z} \cos{\left(x \right)} \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{\partial}{\partial x} x y z + \frac{d}{d z} \sin{\left(y \right)} \right|_{\substack{ x=1\\ y=1\\ z=1 }}\\\left. - \frac{d}{d y} \sin{\left(y \right)} + \frac{d}{d x} \cos{\left(x \right)} \right|_{\substack{ x=1\\ y=1\\ z=1 }}\end{matrix}\right] = \left[\begin{matrix}1\\-1\\- \sin{\left(1 \right)} - \cos{\left(1 \right)}\end{matrix}\right] \]curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = 1,\quad \nabla\times\mathbf F = \left\langle 1, -1, - \sin{\left(1 \right)} - \cos{\left(1 \right)} \right\rangle \)
✓ 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
| 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-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 2026-10-17 with SymPy 1.14.0.