stylistic:style_transfer
Length of a curve

If X = R n {\displaystyle X=\mathbb {R} ^{n}} is the n {\displaystyle n} -dimensional Euclidean space, and if γ: → R n {\displaystyle \gamma :\to \mathbb {R} ^{n}} is an injective and continuously differentiable function, then the length of γ {\displaystyle \gamma } is defined as the quantity

The length of a curve is independent of the parametrization γ {\displaystyle \gamma }.

In particular, the length s {\displaystyle s} of the graph of a continuously differentiable function y = f (x) {\displaystyle y=f(x)} defined on a closed interval {\displaystyle } is

which can be thought of intuitively as using the Pythagorean theorem at the infinitesimal scale continuously over the full length of the curve.

More generally, if X {\displaystyle X} is a metric space with metric d {\displaystyle d}, then we can define the length of a curve γ: → X {\displaystyle \gamma :\to X} by

where the supremum is taken over all n ∈ N {\displaystyle n\in \mathbb {N} } and all partitions t 0 < t 1 < … < t n {\displaystyle t_{0}<t_{1}<\ldots <t_{n}} of {\displaystyle }.

A rectifiable curve is a curve with finite length. A curve γ: → X {\displaystyle \gamma :\to X} is called natural (or unit-speed or parametrized by arc length) if for any t 1, t 2 ∈ {\displaystyle t_{1},t_{2}\in } such that t 1 ≤ t 2 {\displaystyle t_{1}\leq t_{2}}, we have

If γ: → X {\displaystyle \gamma :\to X} is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of γ {\displaystyle \gamma } at t ∈ {\displaystyle t\in } as

and then show that