Line integrals and work
Problem 12.15 · medium
Find the work done by \( \displaystyle \mathbf{F} = \langle x + y, x - y \rangle \) along the unit circle, counterclockwise.
- Parametrize: r(t) = ⟨cos(t), sin(t)⟩ for 0 ≤ t ≤ 2*pi.
- \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(t \right)}\\\frac{d}{d t} \sin{\left(t \right)}\end{matrix}\right] = \left[\begin{matrix}- \sin{\left(t \right)}\\\cos{\left(t \right)}\end{matrix}\right] \]r'(t).✓ Proved
- \[ \left(- \sin{\left(t \right)} + \cos{\left(t \right)}\right) \cos{\left(t \right)} - \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) \sin{\left(t \right)} = \sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)} \]F(r(t)) · r'(t).✓ Proved
- \[ \int\limits_{0}^{2 \pi} \sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)}\, dt = 0 \]W = ∫ F · dr.✓ Proved
Answer \( 0 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | a 20,000-chord polygon along the path gives the same work |
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/line_integral_work, checked 2026-09-26 with SymPy 1.14.0.