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Conservative fields and potential functions

Problem 12.192 · medium

Show that \( \displaystyle \mathbf F = \left(\cos{\left(y \right)}\right)\mathbf i + \left(- x \sin{\left(y \right)}\right)\mathbf j \) is conservative, find a potential function \( \displaystyle f \), and evaluate \( \displaystyle \int_C \mathbf F \cdot d\mathbf r \) along any path from \( \displaystyle (-1, -1) \) to \( \displaystyle (-1, 0) \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(- x \sin{\left(y \right)}\right)\\\frac{d}{d y} \cos{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(y \right)}\\- \sin{\left(y \right)}\end{matrix}\right] \]
∂Q/∂x = ∂P/∂y: F is conservative.✓ Proved
Integrate the first component in x, then fix the 'constant' (a function of the other variables) by matching the other components.
Reviewed
\[ \left[\begin{matrix}\frac{\partial}{\partial x} x \cos{\left(y \right)}\\\frac{\partial}{\partial y} x \cos{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}\cos{\left(y \right)}\\- x \sin{\left(y \right)}\end{matrix}\right] \]
f = x*cos(y) has gradient F.✓ Proved
\[ -1 + \cos{\left(1 \right)} \]
∫_C F·dr = f(end) − f(start), for any path.✓ Proved

Answer \( f = x \cos{\left(y \right)} + C,\quad \int_C \mathbf F\cdot d\mathbf r = -1 + \cos{\left(1 \right)} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	the line integral computed numerically along two different paths

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the vector field as conservative, finds the potential function, and applies the Fundamental Theorem of Line Integrals with correct arithmetic.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the vector field as conservative, finds the potential function, and applies the Fundamental Theorem of Line Integrals with correct arithmetic.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the vector field as conservative, finds the potential function, and applies the Fundamental Theorem of Line Integrals with the correct evaluation.
gpt-oss:20b: pass 2026-10-05

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/conservative_fields, checked 2026-10-05 with SymPy 1.14.0.