Reparametrization invariance of a vector line integral ($\int \mathbf{F}(\gamma (h(s)))\cdot {\gamma }^{\prime }(h(s))h^{\prime }(s)ds=\int \mathbf{F}(\gamma (t))\cdot {\gamma }^{\prime }(t)dt$)

If $h:[\alpha ,\beta ]\to [a,b]$ is an increasing piecewise smooth change of parameter, with endpoints mapped to endpoints, the chain rule gives $(\gamma \circ h{)}^{\prime }=({\gamma }^{\prime }\circ h)h^{\prime }$. Change of variables in the defining line integral proves the displayed identity. A decreasing change of parameter reverses the endpoints and the sign. The result concerns the same traversal with the same multiplicity; retracing a loop additional times changes the integral in general.