Type: Article
Publication Date: 1973-08-01
Citations: 29
DOI: https://doi.org/10.2140/pjm.1973.47.427
If / is a mapping from an open k-cnbe in Rk into R n , 2 5Ξ k ^ n, whose coordinate functions belong to appropriate Sobolev spaces, then the area of / is the integral with respect to k dimensional Hausdorff measure over R n of a nonnegative integer valued multiplicity function.is a mapping whose coordinate functions belong to appropriate Sobolev classes, it was shown in [6] that / is Jf c -1 continuous and that the area of /, as defined in [5], is equal to the classical Jacobian integral.The purpose of this paper is to investigate, using the theory of currents as in [2], the geometric-measure theoretic properties of such a surface and to show that the area is equal to the integral with respect to k dimensional Hausdorff measure in R n of an integer valued multiplicity function.2 Suppose k and n are integers with 2 ^ k ^ n.