Home›Calculus 3›Divergence and curl at a point›Problem 12.348

Divergence and curl at a point

Problem 12.348 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x + y, y + z, x + z \right\rangle \) at \( \displaystyle (2, -1, 3) \).

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

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

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the definitions of divergence and curl. The algebraic derivations are verified as correct, and the evaluation at the specified point is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the definitions of divergence and curl. The algebraic derivations are verified as correct, and the evaluation at the specified point is accurate.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the definitions of divergence and curl. The algebraic substitutions and evaluations at the specified point are accurate.
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 2026-10-17 with SymPy 1.14.0.