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单词 RiemannMultipleIntegral
释义

Riemann multiple integral


We are going to extend the conceptMathworldPlanetmath of Riemann integral to functions of several variables.

Let f:n be a bounded function with compact support.Recalling the definitions of polyrectangle and the definitions of upper and lower Riemann sums on polyrectangles,we define

S*(f):=inf{S*(f,P):P is a polyrectangle, f(x)=0 for every xnP},
S*(f):=sup{S*(f,P):P is a polyrectangle, f(x)=0 for every xnP}.

If S*(f)=S*(f) we say that f is Riemann-integrable on n and we define the Riemann integral of f:

f(x)𝑑x:=S*(f)=S*(f).

Clearly one has S*(f,P)S*(f,P). Also one has S*(f,P)S*(f,P) when P and P are any two polyrectangles containing the supportMathworldPlanetmath of f. In fact one can always find a common refinement P′′ of both P and P so that S*(f,P)S*(f,P′′)S*(f,P′′)S*(f,P). So, to prove that a function is Riemann-integrable it is enough to prove that for every ϵ>0 there exists a polyrectangle P such that S*(f,P)-S*(f,P)<ϵ.

Next we are going to define the integral on more general domains. As a byproduct we also define the measureMathworldPlanetmath of sets in n.

Let Dn be a bounded set. We say that D is Riemann measurable ifthe characteristic functionMathworldPlanetmathPlanetmathPlanetmath

χD(x):={1if xD0otherwise

is Riemann measurable on n (as defined above). Moreover we define the Peano-Jordan measure of D as

𝐦𝐞𝐚𝐬(D):=χD(x)𝑑x.

When n=3 the Peano Jordan measure of D is called the volume of D,and when n=2 the Peano Jordan measure of D is called the area of D.

Let now Dn be a Riemann measurable setMathworldPlanetmath and let f:D be a bounded function. We say that f is Riemann measurable if the function f¯:n

f¯(x):={f(x)if xD0otherwise

is Riemann integrable as defined before. In this case we denote with

Df(x)𝑑x:=f¯(x)𝑑x

the Riemann integral of f on D.

TitleRiemann multiple integral
Canonical nameRiemannMultipleIntegral
Date of creation2013-03-22 15:03:34
Last modified on2013-03-22 15:03:34
Ownerpaolini (1187)
Last modified bypaolini (1187)
Numerical id14
Authorpaolini (1187)
Entry typeDefinition
Classificationmsc 26A42
Related topicPolyrectangle
Related topicRiemannIntegral
Related topicIntegral2
Related topicAreaOfPlaneRegion
Related topicDevelopableSurface
Related topicVolumeAsIntegral
Related topicAreaOfPolygon
Related topicMoscowMathematicalPapyrus
Related topicIntegralOverPlaneRegion
DefinesRiemann integrable
DefinesPeano Jordan
Definesmeasurable
Definesarea
Definesvolume
DefinesJordan content
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更新时间:2025/5/4 16:37:56