Real Analysis II

By Prof. Jaikrishnan J   |   IIT Palakkad
Learners enrolled: 943
This is the follow-up course to Real Analysis I. This time we deal with differentiation and integration of functions of several variables. First, we set the stage by studying metric spaces with special emphasis on normed vector spaces. Even here we will encounter several deep theorems like the existence of the completion of metric space, the Arzela—Ascolli theorem as well as the famous Stone—Weierstrass theorem. 

We will then study the derivative as a linear map and prove the famous implicit and inverse function theorems. These theorems will naturally lead on to the definition of a manifold. We will use the language of manifolds to make precise the method of Lagrange multipliers for constrained optimization. 

Finally, we will take an elementary approach to the Lebesgue integral that bypasses the more abstract and set-theoretic construction via measures. We will prove all the famous convergence theorems. We will also briefly see how our elementary construction can also be quickly obtained using the completion theorems we studied in metric spaces. The final theorem of the course is the difficult Jacobi transformation formula commonly known as change of variables for which we will give a geometric proof. 

This course is designed for ambitious undergraduate students as well as beginning graduate students in mathematics. Knowledge of the content of Real Analysis I is assumed as well as content of a basic course in Linear Algebra at the undergraduate level. I will also assume the basics of an undergraduate level course on multivariable calculus as typically done in the first year of BSc./B.Tech.

INTENDED AUDIENCE : Anyone Interested Learners
PRE-REQUISITES    : Real Analysis I, Linear Algebra, Multivariable Calculus
INDUSTRY SUPPORT : DeepMind, Microsoft Research, OpenAI
Course Status : Ongoing
Course Type : Elective
Duration : 12 weeks
Start Date : 26 Jul 2021
End Date : 15 Oct 2021
Exam Date : 24 Oct 2021 IST
Enrollment Ends : 09 Aug 2021
Category :
  • Mathematics
Credit Points : 3
Level : Undergraduate/Postgraduate

Course layout

Week 1:Metric spaces and normed linear spaces 
Week 2:Compactness and completeness in metric spaces 
Week 3:The Arzelà–Ascoli and Stone--Weierstrass theorems 
Week 4:The derivative in several variables as a linear map I
Week 5:The derivative in several variables as a linear map II
Week 6:The inverse and implicit function theorems
Week 7:Manifolds, Tangent spaces and Lagrange multipliers
Week 8:A precis of curves and surfaces
Week 9:The definition of the Lebesgue integral
Week 10:Convergence theorems for the Lebesgue integral
Week 11:Multiple Lebesgue Integrals
Week 12:The Jacobi transformation formula for Lebesgue integrals

Books and references

  • Mathematical Analysis by Tom Apostol (2nd edition) 
  • Undergraduate Analysis by Serge Lang 
  • Multivariable Mathematics by Ted Shifrin

Instructor bio

Prof. Jaikrishnan J

IIT Palakkad
Jaikrishnan J is an Assistant Professor at IIT Palakkad. He specializes in Complex Analysis.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 24 October 2021 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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