Exercise 6.19

Question

Exercise 19: Let $\gamma_1$ be a curve in $\mathbf R^k$, defined on $[a,b]$, let $\phi$ be a continuous one-to-one mapping of $[c,d]$ onto $[a,b]$ such that $\phi(c)=a$, and
    define $\gamma_2(s)=\gamma_1\bigl(\phi(s)\bigr)$. Prove that $\gamma_2$ is an arc, a closed curve, or a rectifiable curve if and only if the same is true of $\gamma_1$. Prove
    that $\gamma_2$ and $\gamma_1$ have the same length.

analambanomenos · Accepted Answer

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Suppose $\gamma_1$ is an arc. Since the composition of one-to-one mappings is clearly one-to-one, $\gamma_2$ is also an arc.

Note that $\phi$ is strictly monotonically increasing. For if not, there are points $c<x_1<x_2<d$ such that $\phi(c)=a<\phi(x_2)<\phi(x_1)$. by the intermediate-value theorem
    (Theorem 4.23) there is a point $x_3$, $c<x_3<x_1$ such that $\phi(x_3)=\phi(x_2)$, contradicting the fact that $\phi$ is one-to-one. Hence $\phi(d)=b$, so if $\gamma_1$ is 
    a closed curve, then $\gamma_2(d)=\gamma_1\bigl(\phi(d)\bigr)=\gamma_1(b)=\gamma_1(a)=\gamma_1\bigl(\phi(c)\bigr)=\gamma_2(c)$, so that $\gamma_2$ is also closed.

Suppose that $\gamma_1$ is rectifiable. Let $P=\{x_0=a<x_2<\cdots<x_n=c\}$ be a partition of $[a,b]$. Then since $\phi$ is strictly monotonically increasing, $\phi^{-1}(P)$ is
    a partition of $[c,d]$, and we have
    \begin{align*}
        \Lambda\bigl(\phi^{-1}(P),\gamma_2\bigr) &= \sum_{i=1}^n\Big|\gamma_2\bigl(\phi^{-1}(x_i)\bigr)-\gamma_2\bigl(\phi^{-1}(x_{i-1})\bigr)\Big| \
        &= \sum_{i=1}^n\big|\gamma_1(x_i)-\gamma_1(x_{i-1})\big| \
        &= \Lambda(P,\gamma_1).
    \end{align*}
    Hence $\gamma_2$ is rectifiable if and only if $\gamma_1$ is rectifiable, in which case they have the same length.

Exercise 6.19

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Exercise 6.19

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