The multidimensional fundamental theorem of calculus. (English) Zbl 0638.26011

If \(E\subset R^ m\), \(\eta >0\) and \(U(E,\eta)=\{x\in R^ m:\) \(dist(x,E)<\eta \}\), E is said to be thin (resp. slight) if it is compact and \(U(E,\eta)=O(\eta)\) (resp. \(=o(\eta))\) as \(\eta\to 0\). A bounded subset of \(R^ m\) whose boundary is thin is called admissible. The present paper proposes a Riemann type integration for functions defined on admissible sets of \(R^ n\), which coincides with Lebesgue integral for positive functions, provides a Stokes theorem for differentiable vector fields and admits a transformation theorem.
Reviewer: J.Mawhin


26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
26B15 Integration of real functions of several variables: length, area, volume
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
58C35 Integration on manifolds; measures on manifolds