\[\begin{array} {rcl} {\int_{\gamma} F \cdot dr} & = & {\int_a^b F (\gamma (t)) \cdot y' (t)\ dt} \\ {} & = & {\int_a^b \dfrac{df(\gamma (t))}{dt} dt} \\ {} & = & {f(\gamma (b)) - f(\gamma (a))} \\ {}...∫γF⋅dr=∫baF(γ(t))⋅y′(t)dt=∫badf(γ(t))dtdt=f(γ(b))−f(γ(a))=f(P)−f(Q) For a vector field F, the line integral ∫F⋅dr is called path independent if, for any two points P and Q, the line integral has the same value for every path between P and Q.
