Journal of Mathematical Sciences

, Volume 243, Issue 6, pp 917–921

# The Hausdorff Measure on n-Dimensional Manifolds in ℝm and n-Dimensional Variations

Article

We extend the notion of the variation Vf([a; b]) of a function f : [a; b] →  to the variation Vf(A) of a continuous map f : G → n, where G is an open subset of ℝn, over a set AG of the form A = ∪i ∈ IKi where I is countable and all Ki are compact.

Let f : G → m where G ⊂ ℝn with nm, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < inm, let fα be the map with coordinate functions $${f}_{i_1},\dots, {f}_{i_n}$$. The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝn, and a subset AG has the form A = ∪i ∈ IKi where I is countable and all Ki are compact, then $${V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right)$$ where $${V}_{f_{\alpha }}(A)$$ is the variation of fα over A and Hn is the n-dimensional Hausdorff measure in ℝm.

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