Curve - Lengths of Curves

Lengths of Curves

If is a metric space with metric, then we can define the length of a curve by

where the sup is over all and all partitions of .

A rectifiable curve is a curve with finite length. A parametrization of is called natural (or unit speed or parametrised by arc length) if for any, in, we have

If is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of at as

and then

In particular, if is an Euclidean space and is differentiable then

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