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Decomposition of variational measure and the arc-length of a curve in \(\mathbb R^ n\). (English) Zbl 1080.28004

For curves \(\{(f_1(t), \dots, f_n(t));\; t\in [0,c]\}\) in \(\mathbb R^n\) with \(f_i\) continuous and of bounded variation their arc-length is described by a formula which uses integration of the absolute upper \(s\)-derivatives of the components \(f_i\) with respect to the Hausdorff measure \(\mathcal H^s\), \(0< s \leq 1\).

MSC:

28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
26A45 Functions of bounded variation, generalizations
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