Course11110615374 lectures
Real Analysis II
Prof. Jaikrishnan J · IIT Palakkad
About
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
Prerequisites: Real Analysis I, Linear Algebra, Multivariable Calculus
Lectures
Week 1
- 1.2 Metric Spaces
- 1.3 Examples of metric spaces
- 1.4 Loads of definitions
- 2.1 Normed vector spaces
- 2.2 Examples of normed vector spaces
- 2.3 Basic properties open closed sets metric
- 3.1 Continuity in metric spaces
- 3.2 Equivalent metrics and product spaces
Week 2
- 4.1 Completeness
- 4.2 Completeness continued
- 4.3 Completeness of B(x,y)
- 5.1 Completion
- 5.2 Compactness
- 6.1 The Bolzano--Weierstrass Property
- 6.2 Open covers and Compactness
- 6.3 The Heine--Borel Theorem for Metric Spaces
Week 3
- 7.1 Connectedness
- 7.2 Path-Connectedness
- 7.3 Connected Components
- 8.1 The Arzela--Ascolli theorem
- 8.2 Upper and lower limits
- 9.1 The Stone--Weierstrass theorem
- 9.2 All norms are equivalent
Week 4
- 10.1 Vector-valued functions
- 10.2 Scalar-valued functions of a vector variable
- 10.3 Directional derivatives and the gradient
- 11.1 Interpretation and properties of the gradient
- 11.2 Higher-order partial derivatives
- 12.1 The derivative as a linear map
- 12.2 Examples of differentiation
Week 5
- 13.1 Properties of the derivative map
- 13.2 The mean-value theorem
- 13.3 Differentiating under the integral sign
- 14.1 Higher-order derivatives
- 14.2 Symmetry of the second derivative
- 15.1 Taylor's theorem
- 15.2 Taylor's theorem with remainder
Week 6
- 16.1 The Banach fixed point theorem
- 16.2 Newton's method
- 17.1 The inverse function theorem
- 18.1 Diffeomorphismsm and local diffeomorphisms
- 18.2 The implicit function theorem
Week 7
- 19.1 Tangent space to a hypersurface
- 20.1 The definition of a manifold
- 21.1 Examples and non examples of manifolds
- 21.2 The tangent space to a manifold
Week 8
- 22.1 Maxima and minima in several variables
- 22.2 The Hessian and extrema
- 22.3 Completing the squares
- 22.4 Constrained extrema and lagrange multipliers
- 23.1 Curves
- 24.1 Rectifiability and arc-length
Week 9
- 25.1 The Riemann integral revisited
- 25.2 Monotone sequences of functions
- 26.1 Upper functions and their integrals
- 26.2 Riemann integrable functions as upper functions
- 27.1 Lebesgue integrable functions
- 27.2 Approximation of Lebesgure integrable functions
Week 10
- 28.1 Levi monotone convergence theorem for step functions
- 28.2 Monotone convergence theorem for upper functions
- 28.3 Monotone convergence theorem for Lebesgue integrable functions
- 29.1 The Lebesgue dominated convergence theorem
- 29.2 Applications of the convergence theorems
- 30.1 The problem of measure
Week 11
- 31.1. The Lebesgue integral on unbounded intervals
- 31.2 Measurable functions
- 32.1 Solution to the problem of measure
- 33.1 The Lebesgue integral on arbitrary subsets
- 33.2 Square integrable functions
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Palakkad
- Department
- Mathematics
- Teacher
- Prof. Jaikrishnan J
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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