 Reparametrization invariance of a vector line integral

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.