= Solution
{c}
An orientation-preserving <reparametrization> has the form $\widetilde\gamma(s)=\gamma(h(s))$, where $h:[\alpha,\beta]\to[a,b]$ is an increasing, piecewise smooth change of parameter with $h(\alpha)=a$, $h(\beta)=b$. For regular parametrizations one may take $h$ and its inverse continuously differentiable on each piece. The <chain rule> gives $\widetilde\gamma'(s)=\gamma'(h(s))h'(s)$. The <change of variables> $t=h(s)$ therefore yields
$$
\begin{aligned}
\int_\alpha^\beta\mathbf F(\widetilde\gamma(s))\cdot\widetilde\gamma'(s)\,ds
&=\int_\alpha^\beta\mathbf F(\gamma(h(s)))\cdot\gamma'(h(s))h'(s)\,ds\\
&=\int_a^b\mathbf F(\gamma(t))\cdot\gamma'(t)\,dt.
\end{aligned}
$$
This proves <reparametrization invariance of a vector line integral>. If $h$ is decreasing, the endpoints in the final integral are reversed and the sign changes. Thus the same direction and the same multiplicity of traversal are essential.
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