Abstract
It is well known that any compactly supported continuous complex differential n-form can be integrated over real n-dimensional C 1 manifolds in Cm (m ≥ n). For n = 1, the integral along any locally rectifiable curve is defined. Another generalization is the theory of currents (linear functionals on the space of compactly supported C ∞ differential forms). The topic of the article is the integration of measurable complex differential (n, 0)-forms (containing no \(d{\bar z_j}\)) over real n-dimensional C 0 manifolds in Cm with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimensions n > 1). The last result is that a real n-dimensional manifold C 1 embedded in Cm has locally finite variations, and the integral of a measurable complex differential (n, 0)-form defined in the article can be calculated by a well-known formula.
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References
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Original Russian Text © A.V. Potepun, 2016, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2016, No. 1, pp. 44–58.
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Potepun, A.V. Complex vector measure and integral over manifolds with locally finite variations. Vestnik St.Petersb. Univ.Math. 49, 34–46 (2016). https://doi.org/10.3103/S1063454116010118
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DOI: https://doi.org/10.3103/S1063454116010118