Real Analysis part 4 (Riemann Stieltjes Integral)

Name: Real Analysis part 4 (Riemann Stieltjes Integral)
Rating: 4.5 (3 reviews)

Riemann Stieltjes Integral, Riemann, Stieltjes integral, Riemann sums, Riemann, Integrals, Riemann integrals

Created byJaswinder Kaur

Last updated 10/2024

English

What you'll learn

The Real Analysis is a very important and vast branch of Mathematics, applied in Higher Studies, and Riemann Stieltjes Integral is one of its content
This course serves as an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continous probability.
It also serves as an instructive and useful precursor of the Lebesgue Integral.
Students will learn that Riemann sums give us the systematic way to find the area of a curved surface when we know the mathematical function for that curve.

Course content

6 sections • 37 lectures • 8h 6m total length

Riemann Stieltjes Integral Introduction23:01
Riemann Stieltjes Integral Continued....9:25
Explore the Riemann-Stieltjes integral on [a,b] for a monotone increasing alpha, bounded on interval, and construct upper and lower sums via partitions and delta alpha to determine when they coincide.
Riemann Stieltjes Integral Assignment 110:03
Riemann Stieltjes Integral Assignment 213:34
Riemann Stieltjes Integral Assignment 39:52
Riemann Stieltjes Integral Assignment 422:03
Prove that for every epsilon, there exists a partition with the upper minus lower sums of the Riemann–Stieltjes integral less than epsilon, and that this holds for all refinements.
Riemann Stieltjes Integral Assignment 55:32
Show that for a bounded f on [a,b], the sum of |f(s_i)-f(t_i)| delta alpha_i is bounded by upf alpha minus lpf alpha, which is less than epsilon.
Riemann Stieltjes Integral Assignment 614:02
This lecture proves that a continuous f on [a,b], with alpha increasing, is Riemann-Stieltjes integrable on [a,b] using uniform continuity and epsilon-delta arguments.
Riemann Stieltjes Integral Assignment 7(Expected)7:02
Riemann Stieltjes Integral Assignment 816:50
the lecture proves that a bounded f with finitely many discontinuities on [a,b], with alpha continuous at those points, is Riemann integrable, using partitions to bound upper and lower sums.
Riemann Stieltjes Integral Assignment 918:04

Riemann Stieltjes Integral Assignment 1413:34
Riemann Stieltjes Integral Assignment 1512:50
If F1 and F2 are Riemann integrable on [a,b] and F1(x) <= F2(x), then the integral of F1 over [a,b] is no larger than that of F2. It covers subintervals.
Riemann Stieltjes Integral Assignment 1610:14
Riemann Stieltjes Integral Assignment 1720:14
Shows that, for increasing alpha on [a,b], f is Riemann-Stieltjes integrable with respect to alpha iff f is integrable against alpha', and integrals equal ∫ f(x) alpha'(x) dx.
Riemann Stieltjes Integral Assignment 1811:32
Master the change of variables in the Riemann Stieltjes integral by defining beta from alpha and g from f, and proving integral equality under the transformation.

Integration and Differentiation Assignment 122:19
The lecture shows that F(x)=∫_a^x f(t) dt is continuous on [a,b]. It proves that if f is continuous at x0, then F is differentiable at x0 with F'(x0)=f(x0).
Fundamental Theorem of Calculus Assignment 27:58
Integration by parts Assignment 38:50
Derive the integration by parts formula for differentiable F and G on [a, b], showing ∫_a^b F G' = F(b)G(b) − F(a)G(a) − ∫_a^b F' G.
The use of Greatest Integer Function Assignment 412:24
Assignment 516:37

Riemann Steiltjes Integral Examples Assignment 112:14
Revisit the criteria for the Riemann integral on closed intervals using upper and lower sums, partitions, and epsilon bounds, including the irrational–rational example that is not integrable.
Riemann Steiltjes Integral Examples Assignment 210:55
Riemann Steiltjes Integral Examples Assignment 313:16
Riemann Steiltjes Integral Examples Assignment 48:12
Showcases calculating lower and upper integrals for a piecewise f on [0,1], with f equals sqrt(1-x^2) on rationals and 1-x on irrationals; prove not integrable since 1/2 ≠ pi/4.
Riemann Steiltjes Integral Examples Assignment 514:33
Riemann Steiltjes Integral Examples Assignment 614:05
Riemann Steiltjes Integral Examples Assignment 714:19
Riemann Steiltjes Integral Examples Assignment 810:30
This lecture derives two Riemann-Stieltjes identities from assignment 8 by converting sums to integrals and evaluating a log term and related x-integrals.
Riemann Steiltjes Integral Examples Assignment 918:30
Riemann Steiltjes Integral Examples Assignment 1010:25
Explore the Riemann–Stieltjes integral through assignment 10, proving the relation between a discrete sum and a Riemann–Stieltjes integral using summation formulas and integration by parts.

Requirements

The only pre-requisite is a knowledge of Calculus at high school level so that students of science and engineering who want deeper understanding of calculus or want to pursue subjects such as Theoretical Physics, Computational complexity, Statistics etc. would benefit from this course.

Description

Riemann Stieltjes Integral is a generalization of the Riemann Integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. It serves as an instructive and useful precursor of the Lebesgue Integral. and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.

The definition of Riemann Stieltjes Integral uses a sequence of partitions P of the given interval [a,b].The integral, the, is defined to be the limit, as the norm (the length of the longest sub interval) of the partitions

The course introduces the definition and existence of the Riemann Stieltjes integral including the concepts of the following:

Common Refinement of a partition , Upper Riemann and Lower Riemann sum, Riemann Sum, Properties of Riemann Stieltjes Integral, Integration and Differentiation, The Fundamental Theorem Of Calculus,Integration Of Vector Valued Functions,

Rectifiable Curves,that also covers the Expected Theorems and Solved Examples based above contents.

This course also describes the various identities of Riemann Stieltjes integral.

Overall , every thing has been covered with lots of concepts and solved examples , assignments, exercises and almost all the expected theorems.

For any queries related to the course, I would be happy to assist you. Just ping me via Inbox. You will get a course completion certificate after finishing the course.

Who this course is for:

BSc. Graduates , Masters in Sciences( Mathematics), CSIR UGC NET and other Entrances Exams.

Real Analysis part 4 (Riemann Stieltjes Integral)

What you'll learn

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Course content

Introduction of Riemann Steiltjes Integral11 lectures • 2hr 29min

Properties of Riemann Stieltjes Integral4 lectures • 34min

Riemann Stieltjes Sum5 lectures • 1hr 8min

Riemann Stieltjes Integral Integration and Differentiation)5 lectures • 1hr 8min

Integration of Vector Valued Functions2 lectures • 39min

Riemann Steiltjes Integral Solved Examples10 lectures • 2hr 7min

Requirements

Description

Who this course is for: