Higher dimensional mappings for which the area formula holds.
For each continuous mapping of 2 space into n space, n >/= 2, the Lebesgue area is given by the classical formula provided that the partial derivatives exist almost everywhere and belong to the class L(2). The analogous question for mappings of m space into n space, 2 < m </= n, has been open for a long time. We answer this question in the affirmative in a more general setting. Accordingly, as a special case, we show that if a continuous mapping of m space into n space, m </= n, has partial derivatives which belong to L(m) then the Lebesgue area is given by the classical formula.
PURDUE UNIVERSITY, LAFAYETTE, INDIANA.
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Goffman
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Ziemer