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
Read more about this topic: Curve
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