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length of curve in a metric space
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(Definition)
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Suppose that $(X,d)$ is a metric space. Let $f$ be a curve, so that $f: [0,1] \to X$ is a continuous function, and let $0=t_0 < t_1 < \cdots < t_n=1$ and $x_i = f(t_i)$ for $0 \le i \le n$ The set $\{x_0, x_1, \ldots , x_n\}$ is called a partition of the curve. The length of the curve is defined to be the supremum over all partitions of the quantity $\sum_{i=1}^n d(x_i , x_{i-1})$
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"length of curve in a metric space" is owned by Mathprof.
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length of a curve |
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Cross-references: supremum, partition, continuous function, curve, metric space
There are 3 references to this entry.
This is version 5 of length of curve in a metric space, born on 2007-03-16, modified 2007-07-21.
Object id is 9085, canonical name is LengthOfCurveInAMetricSpace.
Accessed 1779 times total.
Classification:
| AMS MSC: | 26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume) |
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Pending Errata and Addenda
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