Home›Calculus 3›Divergence and curl›Problem 12.141

Divergence and curl

Problem 12.141 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 2 x^{2}, - 3 y z, \cos{\left(z \right)} \rangle \).

\[ \frac{d}{d x} \left(- 2 x^{2}\right) + \frac{\partial}{\partial y} \left(- 3 y z\right) + \frac{d}{d z} \cos{\left(z \right)} = - 4 x - 3 z - \sin{\left(z \right)} \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{\partial}{\partial z} \left(- 3 y z\right) + \frac{d}{d y} \cos{\left(z \right)}\\\frac{d}{d z} \left(- 2 x^{2}\right) - \frac{d}{d x} \cos{\left(z \right)}\\- \frac{d}{d y} \left(- 2 x^{2}\right) + \frac{\partial}{\partial x} \left(- 3 y z\right)\end{matrix}\right] = \left[\begin{matrix}3 y\\0\\0\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = - 4 x - 3 z - \sin{\left(z \right)},\quad \nabla\times\mathbf{F} = \langle 3 y, 0, 0 \rangle \)

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	difference quotients of each component at (0.3, 0.7, 1.1) agree

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The divergence calculation is incorrect. The term Derivative(-3*y*z, y) evaluates to -3z, but the stated answer includes -3z in the divergence, implying the term was treated as -3z^2 or similar, or the derivative of cos(z) with respect to z was incorrectly calculated as -sin(z) (which is correct) but the sum is wrong? Wait. Let's re-evaluate. div F = d/dx(-2x^2) + d/dy(-3yz) + d/dz(cos(z)) = -4x - 3z - sin(z). The equation line says the result is -4x - 3z - sin(z). This matches the stated answer. So the divergence is actually correct. Let's re-read the prompt's stated answer. Stated answer: -4x - 3z - sin(z). My calculation: -4x - 3z - sin(z). They match. Why did I think it was wrong? Ah, I misread the vector field or the derivative. d/dy(-3yz) = -3z. Correct. d/dz(cos(z)) = -sin(z). Correct. So divergence is correct. Curl: i component: d/dy(cos(z)) - d/dz(-3yz) = 0 - (-3y) = 3y. j component: d/dz(-2x^2) - d/dx(cos(z)) = 0 - 0 = 0. k component: d/dx(-3yz) - d/dy(-2x^2) = 0 - 0 = 0. Curl is <3y, 0, 0>. This also matches. The solution lines are marked 'proved' by a CAS. The sentences are just labels. There are no sentences to critique for logical flow, only the labels. The labels are correct. The math is correct. Why would this fail? It shouldn't. Wait, look at the divergence again. F = <-2x^2, -3yz, cos(z)>. div F = -4x - 3z - sin(z). The stated answer is -4x - 3z - sin(z). The equation line proves this. The curl is <3y, 0, 0>. The equation line proves this. Everything is correct. I will pass it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The divergence calculation is incorrect. The term Derivative(-3*y*z, y) evaluates to -3z, but the stated answer includes -3z in the divergence, implying the term was treated as -3z^2 or similar, or the derivative of cos(z) with respect to z was incorrectly calculated as -sin(z) (which is correct) but the sum is wrong? Wait. Let's re-evaluate. div F = d/dx(-2x^2) + d/dy(-3yz) + d/dz(cos(z)) = -4x - 3z - sin(z). The equation line says the result is -4x - 3z - sin(z). This matches the stated answer. So the divergence is actually correct. Let's re-read the prompt's stated answer. Stated answer: -4x - 3z - sin(z). My calculation: -4x - 3z - sin(z). They match. Why did I think it was wrong? Ah, I misread the vector field or the derivative. d/dy(-3yz) = -3z. Correct. d/dz(cos(z)) = -sin(z). Correct. So divergence is correct. Curl: i component: d/dy(cos(z)) - d/dz(-3yz) = 0 - (-3y) = 3y. j component: d/dz(-2x^2) - d/dx(cos(z)) = 0 - 0 = 0. k component: d/dx(-3yz) - d/dy(-2x^2) = 0 - 0 = 0. Curl is <3y, 0, 0>. This also matches. The solution lines are marked 'proved' by a CAS. The sentences are just labels. There are no sentences to critique for logical flow, only the labels. The labels are correct. The math is correct. Why would this fail? It shouldn't. Wait, look at the divergence again. F = <-2x^2, -3yz, cos(z)>. div F = -4x - 3z - sin(z). The stated answer is -4x - 3z - sin(z). The equation line proves this. The curl is <3y, 0, 0>. The equation line proves this. Everything is correct. I will pass it.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl using standard formulas. The algebraic results are verified as correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the divergence and curl of the given vector field. The algebraic steps are verified, and the final results match the stated answer.
gpt-oss:20b: pass 2026-09-29

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/divergence_curl, checked 2026-10-04 with SymPy 1.14.0.