Divergence and curl at a point
Problem 12.390 · 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 \( \displaystyle (2, -1, 3) \).
- \[ \left. \frac{\partial}{\partial z} x z^{2} + \frac{\partial}{\partial x} x^{2} y + \frac{\partial}{\partial y} \left(- 3 y z\right) \right|_{\substack{ x=2\\ y=-1\\ z=3 }} = -1 \]div F = P_x + Q_y + R_z, at the point.✓ Proved
- \[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z^{2} - \frac{\partial}{\partial z} \left(- 3 y z\right) \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\\\left. - \frac{\partial}{\partial x} x z^{2} + \frac{\partial}{\partial z} x^{2} y \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\\\left. - \frac{\partial}{\partial y} x^{2} y + \frac{\partial}{\partial x} \left(- 3 y z\right) \right|_{\substack{ x=2\\ y=-1\\ z=3 }}\end{matrix}\right] = \left[\begin{matrix}-3\\-9\\-4\end{matrix}\right] \]curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved
Answer \( \nabla\cdot\mathbf F = -1,\quad \nabla\times\mathbf F = \left\langle -3, -9, -4 \right\rangle \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are correct.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the definitions of divergence and curl, and the algebraic evaluations at the specified point are accurate.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The curl calculation is incorrect. The second component should be P_z - R_x = z^2 - z^2 = 0, not -9. The third component should be Q_x - P_y = 0 - x^2 = -4, which matches, but the first component R_y - Q_z = 0 - (-3y) = 3y = -3 also matches. However, the second component in the solution is calculated as -Derivative(x*z**2, x) + Derivative(x**2*y, z) = -z^2 + 0 = -9, which is wrong because P_z is z^2, not x*z^2 derivative with respect to x. Wait, P = x^2 y, so P_z = 0. R = x z^2, so R_x = z^2. Thus P_z - R_x = 0 - z^2 = -9. My previous check was wrong. Let's re-verify. P=x^2y, Q=-3yz, R=xz^2. Curl = <R_y - Q_z, P_z - R_x, Q_x - P_y>. R_y = 0. Q_z = -3y. So R_y - Q_z = 3y = -3. Correct. P_z = 0. R_x = z^2. So P_z - R_x = -z^2 = -9. Correct. Q_x = 0. P_y = x^2. So Q_x - P_y = -x^2 = -4. Correct. The solution is actually correct. I will mark it as pass.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.