Rectifiable Curves in Rn

Subjects: Vector Analysis
Links: Functions of Bounded Variation on R, Vector-valued functions of R, Differentiabilty of vector valued functions of R

Def: A continuous function $f:[a,b]\to {\mathbb{R}}^n$ is called a path in ${\mathbb{R}}^n$ from $f(a)$ to $f(b)$. The path is differentiable, ${\mathcal{C}}^{\mathcal{1}}$ or smooth if the function $f$ is respectively differentiable, ${\mathcal{C}}^{\mathcal{1}}$ or smooth. If $f(a)=f(b)$ then it is a closed path.

Def: A path $f:[a,b]\to {\mathbb{R}}^n$ is called simple if $f$ is not closed $f$ is injective, if it is closed $f$ is injective on $[a,b)$ or $(a,b]$.

Prop: Let $f:[a,b]\to {\mathbb{R}}^n$ is a nonsimple path with a finite number of self-intersections, then we can partition it into simple paths.

Def: Let $f:[a,b]\to {\mathbb{R}}^n$, and $\mathcal{P}$ be a partition of $[a,b]$, we can define the length of the polygonal path with vertices as $f(x_0),\dots ,f(x_n)$ , in this order.

The length of $f$ is:

if $\mathrm{\Lambda }(f)<\mathrm{\infty }$, then $f$ is called rectifiable, otherwise is nonrectifiable.

There are two notations: $\mathrm{\Lambda }(f)$ is primarily used when the domain is clear or unchanging, and ${\mathrm{\Lambda }}_f(a,b)$ to make the domain clear or changing

Th: Let $f:[a,b]=I\to {\mathbb{R}}^n$, $f$ is rectifiable iff every component $f_k$ is of bounded variation on $[a,b]$. If $f$ is rectifiable then for any $1\le k\le n$:

Th: Let $f:[a,b]\to {\mathbb{R}}^n$ be rectifiable and $c\in (a,b)$, then:

Th: If $f:[a,b]\to {\mathbb{R}}^n$ and $f\in {\mathcal{C}}^{\mathcal{1}}$, then $f$ is rectifiable, and:

Th: Let $f:[a,b]\to {\mathbb{R}}^n$ and $g:[c,d]\to {\mathbb{R}}^n$ be two $1-1$ continuous parametrizations of the curve $C$, them $\mathrm{\Lambda }(f)=\mathrm{\Lambda }(h)$.

Th: Let $f:[a,b]\to {\mathbb{R}}^n$ and $g:[c,d]\to {\mathbb{R}}^n$ equivalent ${\mathcal{C}}^{\mathcal{1}}$ parametrizations of the curve $C$, them $\mathrm{\Lambda }(f)=\mathrm{\Lambda }(h)$.

Cor: All smooth parametrizations of a smooth simple arc in ${\mathbb{R}}^n$ have the same length.

Def: The ******length $\mathrm{\Lambda }(C)$ of a smooth simple arc in $C$ in ${\mathbb{R}}^n$ is the length of any smooth parametrization of $C$.

Def: A ${\mathcal{C}}^{\mathcal{1}}$ parametrization $\gamma :E\subseteq \mathbb{R}\to {\mathbb{R}}^n$ of a curve $C$ in ${\mathbb{R}}^n$ is a path-length parametrization of $C$ if $\parallel {\gamma }^{\prime }(s)\parallel =1$ for all $s\in E$.

Arc-lenght Parametrization

Def: Let $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be smooth parametrization of a curve $C$ in ${\mathbb{R}}^n$. Choose a base point $p\in D$. Then arc-length function based at $p$ is the function $\lambda :D\subseteq \mathbb{R}\to \mathbb{R}$ defined by:

where $s=\lambda (t)$ represents the distance travelled by the particle in the interval between $p$ and $t$. The speed of $f$ at time given by:

Thus $\lambda$ is invertible.

Th: For any smooth parametrization $f:D\subseteq \mathbb{R}\to {\mathbb{R}}^n$ of a curve $C$ in ${\mathbb{R}}^n$ there’s is a arc-length parametrization $\gamma :E\subseteq \mathbb{R}\to {\mathbb{R}}^n$ of $C$ which is properly equivalent to $f$.

Chose a base point $p\in D$, and let $\lambda :D\to \mathbb{R}$ be arc-length function based at $p$. $E=\lambda (D)$, then $\gamma =f\circ {\lambda }^{-1}:E\subseteq \mathbb{R}\to {\mathbb{R}}^n$ is an equivalent parametrization by arc-length. This process is called re-parametrization by arc-length.

Def: Let $f:[a,b]\to {\mathbb{R}}^n$ be a path. The inverse path is defined to be the function $f^{-}:[a,b]\to {\mathbb{R}}^n$ where:

Concatenation of Paths

Def Let $f:[a,b]\to {\mathbb{R}}^n$ and $g:[c,d]\to {\mathbb{R}}^n$ be paths in ${\mathbb{R}}^n$ where $f(b)=g(c)$. The product or concatenation path $h=fg=f\oplus g=f\ast g=f+g:[a,b+d-c]\to {\mathbb{R}}^n$ is defined by: