Work Along a Path

Calculus 3 · Axiom Academy

A force pushes an object along a curve. How much work does it do — and does the path you take matter? That question is a line integral: W = ∫ C F·dr. A hiker climbs a hill, a charge drifts through a field, wind pushes a boat — in each, a force acts while something moves along a path , and the total work is the sum of force-times-distance along the motion . Three moves make that idea yours. A hiker carries a pack up a hill. Gravity pulls straight down the whole way — drag the summit height and watch the work. Flat steps cost nothing; only the climb does, and a twisty trail costs exactly the same as a straight ramp. Only the part along the motion counts Zoom in on one tiny step. The work is the force projected onto the direction you move : W = |F| |dr| cos . Swing the angle — push straight along and you get it all; push sideways at 90° and gravity, or any force, does nothing . Gravity didn't care how you got there. A swirling field — wind eddies, a magnetic push — does. Same start, same finish; drag the path to bow it outward and watch the work climb. In fields like this, W = ∫ C F·dr depends on the whole route.

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