Riemann-Stieltjes Integral geometric intepretation

In summary: Moreover, i am a little confused on what do they mean when they say f is integrable with respect to the g function. E.g. (integrate f dg). Do they treat g(x) as a function, g(x)= y and then (integrate f (y) dy)?Igor Podlubny wrote a paper on finding a geometric interpretation for the Riemann-Stieltjies integral. He defines it as the area under a vector function. If g is differentiable, then this area is the integral of fdg/dx.
  • #1
Zeato
7
0
Hi all,

I would like to ask for the geometric interpretation of the riemann-stieltjies integral.

Suppose we have an integral, (integrate f dg) over the interval [a,b], where g is monotonically increasing.

Can i interpret it as the area between f and the g function?

Moreover, i am a little confused on what do they mean when they say f is integrable with respect to the g function. E.g. (integrate f dg). Do they treat g(x) as a function, g(x)= y and then (integrate f (y) dy) ?
 
Physics news on Phys.org
  • #2
Igor Podlubny wrote a paper on finding http://arxiv.org/abs/math/0110241" [Broken] This paper relies on a previous result about finding a geometrical interpretation for Riemann-Stieltjies. You can read Podlubny's paper for a short synopsis on that result (it should have a citation in there).
 
Last edited by a moderator:
  • #3
Hi pwsnafu,

Thanks for the article.
Although i am able to understand intuitively from the article that the geometric interpretation for the riemann-stieltjes integral is actually the area under a vector function (E.g. K( f(x), g(x), x) for ( integrate f dg ) )
I am still unable to comprehend how they actually come to such a conclusion. I mean for Riemann, Darboux integral, we often look at the the partition over an interval x, but the article mention the use of projection of vectors instead. Also, where do they take the partition in this case? Do they take the partition from the plane that contain f, g, and x vectors? How do they come to such a conclusion then?

Also, when they say f is integrable with respect to g, how do we interpret this geometrically, with the shadow and the fence in the mentioned article?
 
Last edited:
  • #4
Note that in the Riemann-Stieljes integral, [itex]\int f(x)dg[/itex], g only has to be "monotone increasing". It does not have to be differentiable or even continuous. Taking g(x) to be the "greatest integer less than x" gives
[tex]\int_0^N f(x)dg= \sum_{n=0}^N f(n)[/tex]

If g is, in fact, differentiable, then [itex]\int f(x)dg= \int f(x)g'(x)dx[/itex]
 
  • #5
Hi HallsofIvy,

Thanks for the reply.
But how would we arrive at a geometric interpretation for the Riemann-Stieltjes integral?
Also what does it mean geometrically to say that f is integrable with respect to g?

Thanks in advance. :)
 
  • #6
Zeato said:
Hi all,

Can i interpret it as the area between f and the g function?

I don't think so. The area between g and f is the integral of f-g.
If g has a continuous derivative then this looks like the integral of fdg/dx to me.
 

What is the Riemann-Stieltjes Integral geometric interpretation?

The Riemann-Stieltjes Integral geometric interpretation is a method of calculating the area under a curve using a differentiable function and a set of points on the curve. It is an extension of the Riemann Integral and provides a more general way of calculating integrals.

How is the Riemann-Stieltjes Integral geometric interpretation different from the Riemann Integral?

The Riemann-Stieltjes Integral uses a differentiable function as the integrator, whereas the Riemann Integral uses a constant. This allows for a more general approach to calculating integrals, as the integrator can vary for different points on the curve.

What is the role of the integrator in the Riemann-Stieltjes Integral geometric interpretation?

The integrator in the Riemann-Stieltjes Integral represents the function that is being integrated over. It determines the width of each rectangle used to calculate the area under the curve. A different integrator can lead to a different value for the integral.

How is the Riemann-Stieltjes Integral geometric interpretation used in real-world applications?

The Riemann-Stieltjes Integral is used in many areas of science and engineering to calculate various physical quantities, such as work, center of mass, and moments of inertia. It is also used in economics and finance to calculate prices and returns on investments.

What are the limitations of the Riemann-Stieltjes Integral geometric interpretation?

The Riemann-Stieltjes Integral can only be used for functions that are continuous and have a finite number of discontinuities. It also requires the integrator to be differentiable, which limits its applicability in some cases. Additionally, it can be challenging to calculate the integral for complex functions.

Similar threads

  • Calculus
Replies
6
Views
1K
Replies
1
Views
1K
Replies
20
Views
2K
Replies
16
Views
2K
Replies
1
Views
751
Replies
28
Views
3K
Replies
9
Views
2K
Replies
1
Views
785
Replies
4
Views
1K
Replies
5
Views
1K
Back
Top