Line Integrals of Vector Fields

Calculus 3 · Axiom Academy

LESSON Line Integrals of Vector Fields The work a force field does along a path is the running sum of its component along the motion — and it flips sign when you reverse the path. 1. Work Is the Tangential Component, Accumulated Given a vector field and a smooth oriented curve C , the line integral is the work does moving a particle along C . At each instant only the part of pointing along the motion counts — the component on the unit tangent : work = the tangential component , summed over the path Where leans with the motion the contribution is positive; where it leans against, negative; where it is perpendicular to the motion, it contributes nothing. 2. Breaking It Into Components: Write the field in components. If and , the dot product spreads the integral across the coordinate directions: Each term pairs one component of the field with one direction of travel — P with the horizontal step dx , Q with the vertical step dy (and R with dz in 3D). Take , so P=y and Q=x . The differential form is . Along a curve C from (0,0) to (1,1) the two running pieces each build to , so . 3. Evaluating by Parametrization To actually compute, describe the curve by a parameter: for . Since , everything collapses to an ordinary single-variable integral in t : Substitute the field and the velocity, dot them, and integrate over [a,b] . Sweeping the point along the curve is the same motion as sweeping t across the interval — the line integral becomes an area under one scalar curve.

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