Home›Calculus 3›Divergence and curl at a point›Problem 12.386

Divergence and curl at a point

Problem 12.386 · easy

Find the divergence and the curl of \( \displaystyle \mathbf F = \left\langle x y, y z, x z \right\rangle \) at \( \displaystyle (0, 1, 2) \).

\[ \left. \frac{\partial}{\partial x} x y + \frac{\partial}{\partial z} x z + \frac{\partial}{\partial y} y z \right|_{\substack{ x=0\\ y=1\\ z=2 }} = 3 \]
div F = P_x + Q_y + R_z, at the point.✓ Proved
\[ \left[\begin{matrix}\left. \frac{\partial}{\partial y} x z - \frac{\partial}{\partial z} y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. \frac{\partial}{\partial z} x y - \frac{\partial}{\partial x} x z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\\\left. - \frac{\partial}{\partial y} x y + \frac{\partial}{\partial x} y z \right|_{\substack{ x=0\\ y=1\\ z=2 }}\end{matrix}\right] = \left[\begin{matrix}-1\\-2\\0\end{matrix}\right] \]
curl F = ⟨R_y − Q_z, P_z − R_x, Q_x − P_y⟩, at the point.✓ Proved

Answer \( \nabla\cdot\mathbf F = 3,\quad \nabla\times\mathbf F = \left\langle -1, -2, 0 \right\rangle \)

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: fail (error) — The curl calculation is incorrect. The standard formula for curl is \(\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\). The solution computes the z-component as \(Q_x - P_y\) but implements it as \(-P_y + Q_x\) (which is correct algebraically) but the code shows `Subs(-Derivative(x*y, y) + Derivative(y*z, x), ...)` which evaluates to \(-x + 0 = 0\) at (0,1,2). Wait, let's re-evaluate. \(P=xy, Q=yz, R=xz\). \(Q_x = 0\), \(P_y = x\). So \(Q_x - P_y = -x\). At (0,1,2), this is 0. The solution gets 0. The x-component: \(R_y - Q_z = 0 - y = -y\). At (0,1,2), this is -1. The solution gets -1. The y-component: \(P_z - R_x = 0 - z = -z\). At (0,1,2), this is -2. The solution gets -2. The result \(\langle -1, -2, 0 \rangle\) is actually correct. Why did I think it was wrong? Let me re-read the code. `Subs(Derivative(x*z, y) - Derivative(y*z, z), ...)` -> \(0 - y = -y\). Correct. `Subs(Derivative(x*y, z) - Derivative(x*z, x), ...)` -> \(0 - z = -z\). Correct. `Subs(-Derivative(x*y, y) + Derivative(y*z, x), ...)` -> \(-x + 0 = -x\). Correct. The result is correct. The text says `curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>`. This is the correct formula. The implementation matches. The result matches. So why fail? Ah, look at the stated answer in the problem description: `\nabla\cdot\mathbf F = 3`. Let's check divergence. \(P_x = y\), \(Q_y = z\), \(R_z = x\). Sum = \(y+z+x\). At (0,1,2), sum = \(1+2+0 = 3\). Correct. So the solution is actually correct. I must have hallucinated an error. Let me double check the curl formula signs. \(\nabla \times \mathbf{F} = \det \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix} = \langle R_y - Q_z, -(R_x - P_z), Q_x - P_y \rangle = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\). The solution uses this exact formula. The evaluation is correct. The verdict should be pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The curl calculation is incorrect. The standard formula for curl is \(\nabla \times \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\). The solution computes the z-component as \(Q_x - P_y\) but implements it as \(-P_y + Q_x\) (which is correct algebraically) but the code shows `Subs(-Derivative(x*y, y) + Derivative(y*z, x), ...)` which evaluates to \(-x + 0 = 0\) at (0,1,2). Wait, let's re-evaluate. \(P=xy, Q=yz, R=xz\). \(Q_x = 0\), \(P_y = x\). So \(Q_x - P_y = -x\). At (0,1,2), this is 0. The solution gets 0. The x-component: \(R_y - Q_z = 0 - y = -y\). At (0,1,2), this is -1. The solution gets -1. The y-component: \(P_z - R_x = 0 - z = -z\). At (0,1,2), this is -2. The solution gets -2. The result \(\langle -1, -2, 0 \rangle\) is actually correct. Why did I think it was wrong? Let me re-read the code. `Subs(Derivative(x*z, y) - Derivative(y*z, z), ...)` -> \(0 - y = -y\). Correct. `Subs(Derivative(x*y, z) - Derivative(x*z, x), ...)` -> \(0 - z = -z\). Correct. `Subs(-Derivative(x*y, y) + Derivative(y*z, x), ...)` -> \(-x + 0 = -x\). Correct. The result is correct. The text says `curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>`. This is the correct formula. The implementation matches. The result matches. So why fail? Ah, look at the stated answer in the problem description: `\nabla\cdot\mathbf F = 3`. Let's check divergence. \(P_x = y\), \(Q_y = z\), \(R_z = x\). Sum = \(y+z+x\). At (0,1,2), sum = \(1+2+0 = 3\). Correct. So the solution is actually correct. I must have hallucinated an error. Let me double check the curl formula signs. \(\nabla \times \mathbf{F} = \det \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix} = \langle R_y - Q_z, -(R_x - P_z), Q_x - P_y \rangle = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\). The solution uses this exact formula. The evaluation is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The curl calculation contains sign errors in the component formulas. The standard definition is curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>. The solut
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The divergence calculation is incorrect because it omits the partial derivative of the z-component with respect to z (R_z = x, which is 0 at the point, but the formula shown is incomplete/misleading as it only sums P_x and Q_y explicitly in the text description, though the code might be correct, the text says 'div F = P_x + Q_y + R_z' but the equation only shows two terms being added before substitution? Actually, looking closely at line 1: `Derivative(x*y, x) + Derivative(y*z, y)` is missing `Derivative(x*z, z)`. The text says `+ Derivative(x*z, z)` is part of the sum? No, the equation is `Subs((Derivative(x*y, x) + Derivative(y*z, y)) + Derivative(x*z, z), ...)`. Wait, the parentheses are `(A + B) + C`. So the code includes all three. Let's re-read carefully. `Subs((Derivative(x*y, x) + Derivative(y*z, y)) + Derivative(x*z, z), ...)` includes all three. P_x = y, Q_y = z, R_z = x. At (0,1,2): 1 + 2 + 0 = 3. This is correct. Now check curl. Formula: `<R_y - Q_z, P_z - R_x, Q_x - P_y>`. R = xz, Q = yz, P = xy. R_y = 0, Q_z = y. Component 1: 0 - y = -y. At (0,1,2): -1. Correct. P_z = 0, R_x = z. Component 2: 0 - z = -z. At (0,1,2): -2. Correct. Q_x = 0, P_y = x. Component 3: 0 - x = -x. At (0,1,2): 0. Correct. Wait, let's look at the code in line 2. Row 1: `Derivative(x*z, y) - Derivative(y*z, z)`. R_y - Q_z. Correct. Row 2: `Derivative(x*y, z) - Derivative(x*z, x)`. P_z - R_x. Correct. Row 3: `-Derivative(x*y, y) + Derivative(y*z, x)`. -P_y + Q_x. Correct. The solution seems mathematically correct. Why did I think it was wrong? Let me re-read the prompt's instruction. "Find what is wrong". Let's check the divergence text again. `div F = P_x + Q_y + R_z`. The equation is `Subs((Derivative(x*y, x) + Derivative(y*z, y)) + Derivative(x*z, z), ...)`. This evaluates to `y + z + x`. At (0,1,2) this is `1 + 2 + 0 = 3`. Correct. Let's check the curl text again. `curl F = <R_y - Q_z, P_z - R_x, Q_x - P_y>`. Code Row 1: `Derivative(x*z, y) - Derivative(y*z, z)`. `0 - y = -y`. At (0,1,2) -> -1. Correct. Code Row 2: `Derivative(x*y, z) - Derivative(x*z, x)`. `0 - z = -z`. At (0,1,2) -> -2. Correct. Code Row 3: `-Derivative(x*y, y) + Derivative(y*z, x)`. `-x + 0 = -x`. At (0,1,2) -> 0. Correct. Is there a subtle error? Problem: F = <xy, yz, xz>. Div: d/dx(xy) + d/dy(yz) + d/dz(xz) = y + z + x. At (0,1,2): 1 + 2 + 0 = 3. Correct. Curl: i: d/dy(xz) - d/dz(yz) = 0 - y = -y. At (0,1,2): -1. j: d/dz(xy) - d/dx(xz) = 0 - z = -z. At (0,1,2): -2. k: d/dx(yz) - d/dy(xy) = 0 - x = -x. At (0,1,2): 0. Result: <-1, -2, 0>. Correct. The solution is 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.