Consider the field F = (−y, x, 0) (a rotation in the xy-plane). Verify Stokes' theorem for the unit circle in the plane z = 0, traversed anticlockwise. Step 1: Curl: ∇×F = (∂/∂y(0) − ∂/∂z(x), ∂/∂z(−y) − ∂/∂x(0), ∂/∂x(x) − ∂/∂y(−y)) = (0, 0, 1 + 1) = (0, 0, 2). Uniform "whirlpool" of strength 2 pointing along +z. Step 2: Surface integral over the unit disc: ∫(∇×F)·dA = 2 × (area) = 2 × π(1)² = 2π. Step 3: Line integral: on the circle, r = (cosθ, sinθ, 0), dl = (−sinθ, cosθ, 0)dθ. F·dl = (−sinθ)(−sinθ) + (cosθ)(cosθ) dθ = (sin²θ + cos²θ)dθ = dθ. ∮F·dl = ∫₀²π dθ = 2π ✓. Both give 2π: the uniform curl 2 over the disc exactly equals the circulation around the rim. This is the pattern behind ∇×E = −∂B/∂t: magnetic change through a loop drives electric circulation around it.