Generalized N-dimensional Riemann Integral

Let I=[a1,b1]××[aN,bN]N be a compactPlanetmathPlanetmath interval, and let f:IM be a function. Let ϵ>0. If there exists a yM and a partition Pϵ of I such that for each refinementPlanetmathPlanetmath P of Pϵ (and corresponding Riemann SumMathworldPlanetmath S(f,P)),


Then we say that f is Riemann integrablePlanetmathPlanetmath over I, that y is the Riemann integral of f over I, and we write


Note also that it is possible to extend this definition to more arbitrary sets; for any bounded set D, one can find a compact interval I such that DI, and define a function


in which case we define

Title Generalized N-dimensional Riemann Integral
Canonical name GeneralizedNdimensionalRiemannIntegral
Date of creation 2013-03-22 13:37:43
Last modified on 2013-03-22 13:37:43
Owner vernondalhart (2191)
Last modified by vernondalhart (2191)
Numerical id 6
Author vernondalhart (2191)
Entry type Definition
Classification msc 26B12