= Reparametrization invariance of a vector line integral
{title2=$\int\mathbf F(\gamma(h(s)))\cdot\gamma'(h(s))h'(s)ds=\int\mathbf F(\gamma(t))\cdot\gamma'(t)dt$}
If $h:[\alpha,\beta]\to[a,b]$ is an increasing piecewise smooth change of parameter, with endpoints mapped to endpoints, the <chain rule> gives $(\gamma\circ h)'=(\gamma'\circ h)h'$. <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.
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