Work in multiple dimensions

A line integral follows the actual route

In multiple dimensions, the force can change direction and magnitude along a curve. Work is therefore a geometric accumulation along the path, not simply ‘force times distance’ unless the force is constant and parallel.

Parameterize the path

Let r(s) describe a curve and F(r) the force field. The small displacement is dr, so W = ∫C F(r)·dr. For a spring-like field F = (−kx, 0), moving from x₁ to x₂ gives W = ∫{x₁}^{x₂} −kx dx = −½k(x₂² − x₁²), independent of the route because the force is conservative.

For F = (2x, 3y) N and a straight path from (0,0) to (2,1) parameterized r(t) = (2t,t), dr = (2,1)dt. Then W = ∫₀¹ [4t,3t]·[2,1]dt = ∫₀¹ 11t dt = 5.5 J.

Closed-loop work tests conservativeness

A conservative field gives zero work around every closed loop. If a calculated loop integral is nonzero, the field or model includes path-dependent effects such as friction or a nonconservative driving force.

Work it: Evaluate the work of F = (−y,x) around the unit circle counterclockwise. Use r(t) = (cos t, sin t) and interpret the sign physically.