Riemann multiple integral
We are going to extend the concept of Riemann integral to functions of several variables.
Let 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
If we say that is Riemann-integrable on and we define the Riemann integral of :
Clearly one has . Also one has when and are any two polyrectangles containing the support of . In fact one can always find a common refinement of both and so that . So, to prove that a function is Riemann-integrable it is enough to prove that for every there exists a polyrectangle such that .
Next we are going to define the integral on more general domains. As a byproduct we also define the measure of sets in .
Let be a bounded set. We say that is Riemann measurable if the characteristic function
is Riemann measurable on (as defined above). Moreover we define the Peano-Jordan measure of as
When the Peano Jordan measure of is called the volume of , and when the Peano Jordan measure of is called the area of .
Let now be a Riemann measurable set and let be a bounded function. We say that is Riemann measurable if the function
is Riemann integrable as defined before. In this case we denote with
the Riemann integral of on .
Title | Riemann multiple integral |
Canonical name | RiemannMultipleIntegral |
Date of creation | 2013-03-22 15:03:34 |
Last modified on | 2013-03-22 15:03:34 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 14 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 26A42 |
Related topic | Polyrectangle |
Related topic | RiemannIntegral |
Related topic | Integral2 |
Related topic | AreaOfPlaneRegion |
Related topic | DevelopableSurface |
Related topic | VolumeAsIntegral |
Related topic | AreaOfPolygon |
Related topic | MoscowMathematicalPapyrus |
Related topic | IntegralOverPlaneRegion |
Defines | Riemann integrable |
Defines | Peano Jordan |
Defines | measurable |
Defines | area |
Defines | volume |
Defines | Jordan content |