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...Math chapter vector calculus line integrals consider a smooth plane curve c in the space given by parametric equations x t y z b or r i j k then is continuous and construct uniform partition of into n subintervals with point it show fig book for corresponding along arc length parameter s denition integral scalar function f dened on ds lim xf if this limit exists theorem evaluation func tion discuss between can be evaluated as bf dx dy dz dt respect to field e eld that assigns each three dimensional ca expressed p q flow lines streamlines ow are curves such vectors tangents these alternative paths followed particles whose velocity more precisely has representation components satisfy dierential equation work done move particle force w tds where unit tangent at using cf dr why hp hx cp cq cr integration conservative fields domain d containing dierentiable its gradient proof contains means there zcf zc examples independence path two contained have same initial terminal independent dris onl...