UM 102

Scroll down for videos of classes from the twenty-fourth class onward.

Practice problems for mid term. Practice problems - Implicit Differentiation

Practice problems on Tangent planes and normal lines

Assignment 1 (submit by 03022020) Assignment 2 (submit to the TAs at 0930 on 10022020) Assignment 3 (submit by 20032020, delayed due to closure of the institute)

Modified Assignment 3 (Problems 8 and 9 replaced because they were found to be long and difficult, submit Modified Assignment 3 instead of Assignment 3 after the institute opens)

Assignment 4 posted on 05052020 (send scanned copies of solutions of both 3 and 4 to your TA since it is not clear when we shall meet face to face)

Syllabus:

Linear Algebra continued: Inner products and Orthogonality; Determinants; Eigenvalues and Eigenvectors; Diagonalisation of Symmetric matrices.

Multivariable calculus: Functions on Rn ; Partial and Total derivatives; Chain rule; Maxima, minima and saddles; Lagrange multipliers; Integration in Rn, change of variables, Fubini's theorem; Gradient, Divergence and Curl; Line and Surface integrals in R2 and R3 ; Stokes, Green's and Divergence theorems.

Introduction to Ordinary Differential Equations; Linear ODEs and Canonical forms for linear transformations.

My office hours for this course are 5:30 PM to 6:30 PM on Tuesdays and Thursdays.

I taught this course in 2017, 2015 and 2014.

The problem sets of 2015 are here: Exercise Set I , Exercise Set II , Exercise Set III (It says Problem Set V because it was the fifth problem set in 2014. It is Problem Set III for 2015), Exercise Set IV (It says Problem set III because it was the third problem set in 2014. It is problem set IV for 2015.)

Mid Sem Question paper of 2015 is here. The mid sem paper for 2014 is here.

The classes will roughly follow this pattern:

01012020 First class : Revision of basics of vector spaces and Rank-Nullity Theorem. Inner products and Cauchy-Schwarz inequality. Orthogonality. Reference: Apostol's Calculus volume I, sections 15.10 and 16.3.

Documents: Various proofs of the Cauchy-Schwarz inequality

03012020 Second class : Gram-Schmidt process. Parseval's formula, Projections. Reference: Apostol's Calculus volume I, sections 15.11, 15.13 and 15.14, 15.15 and 16.17.

06012020 Third class : Approximation theorem, system of equations. Legendre Polynomials.

Documents: Legendre Polynomials notes from a site at Rochester and closer home a very good set of notes from SERC.

08012020: No class

09012020: Tutorial

10012020: First class test

13012020 Fourth class: Determinants - motivation through cross product, axiomatic definition. I shall mostly follow Chapter 4 of this book: https://textbooks.math.gatech.edu/ila/ila.pdf . Other reference: Apostol's Calculus volume II, most of Chapter 3.

15012020: Holiday

16012020 Fifth class: Existence and uniqueness of the determinant function, computations. Some elementary examples. Reference: Apostol's Calculus volume II, most of Chapter 3.

17012020 Sixth class: Hermitian operators on an inner product space - orthogonality of eigenvectors corresponding to different eigenvalues, orthonormal eigenbasis. Reference: Apostol's Calculus volume II, first half of Chapter 5.

20012020 Seventh class: Diagonalization of hermitian matrices, Schur's upper-triangularization, Cayley-Hamilton Theorem. Reference: Horn and Johnson, Matrix Analysis, second edition, section 2.3.1 and section 5.9 of Apostol's Calculus volume II.

22012020 Eighth class: System of equations, Gauss Jordan elimination, row reduced echelon form. Reference: Linear Algebra by Hoffman and Kunze, Chapter 1.

23012020: Tutorial

24012020 Ninth class: Introduction to calculus of several variables. Reference: Chapter 8 of Apostol's Calculus volume II.

27012020 Tenth class: Calculus of several variables continued.... References: Chapter 8 of Apostol's Calculus volume II, Chapter 3 of Apostol's Mathematical Analysis. Also, equivalence of p-norms in a finite dimensional vector space.

29012020 Eleventh class: Calculus of several variables continued.... Total derivative. References: Sections 8.6, 8.7, 8.8, 8.10 and 8.11 of Apostol's Calculus volume II

30012020: Tutorial

31012020: Classes suspended because of Pravega 2020.

03022020 Twelfth class: Calculus of several variables continued.... Differentiability implies continuity, Gradient, a sufficient condition for differentiability. References: Sections 8.12 and 8.13 of Apostol's Calculus volume II

05022020 Thirteenth class: Chain rule, level sets and tangent planes. References: Sections 8.15 and 8.16 of Apostol's Calculus volume II.

06020202 Fourteenth class: Examples of tangent planes and normal line. Jacobian matrix.

07022020: Tutorial

10022020: Second class test.

12022020: Fifteenth class: Calculus of several variables continued....

14022020: Sixteenth class: Line Integral, potential function. References: Sections 10.1 to 10.5 and 10.11 of Apostol's Calculus volume II.

Mid term examinations and hence classes suspended for about two weeks.

28022020: Seventeenth class: Chain Rule, Implicit differentiation. References: Sections 8.20, 8.21, 8.22, 9.6, 9.7 and 9.8 of Apostol's Calculus volume II.

02032020: Eighteenth class: Development of Taylor formula up to second order, Hessian matrix. References: Section 9.10 of Apostol's Calculus volume II.

04032020: Nineteenth class: Maxima and minima. References: Sections 9.9 of Apostol's Calculus volume II.

06032020: Twentieth class: Maxima and minima continued.

09032020: Twenty-first class: Saddle points.

11032020: Twenty-second class: Multiple integration. References: Sections 11.1 - 11.3, 11.5, 11.6 of Apostol's Calculus volume II.

13032020: Twenty-third class: Boundedness of continuous functions, continuous functions are integrable, Sections 9.16 and 11.10 of Apostol's Calculus volume II.

16032020: Classes suspended till further notice from the institute.

Starting now, i.e., 18 April, 2020, I shall post videos because it is not clear when we shall be able to meet again. These videos will be in two parts, covering roughly one hour in total. It is a pity that the videos do not communicate the energy and enthusiasm that I bring to the class. The material is important and that is the only thing that can be shared. But, other important things like the pleasure of doing mathematics with all of you and interactions with questions can only happen face to face. Well, we can only do what circumstances allow us. We shall try to make the best of circumstances. Looking forward to teaching you again on blackboard in person.

Twenty-fourth class (uploaded on 18042020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Twenty-fifth class (uploaded on 24042020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Twenty-sixth class (uploaded on 28042020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Twenty-seventh class (uploaded on 04052020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Twenty-eighth class (uploaded on 11052020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Twenty-ninth class (uploaded on 25052020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Thirtieth class (uploaded on 27052020): Video lecture: Part 1 and Part 2. Here is the PDF file of notes.

Thirty-first class (uploaded on 01062020): Video lecture: Part 1 and Part 2. Here is a very good set of notes.

Thirty-second class (uploaded on 06062020): Video lecture: Part 1 and Part 2. A good part of the class is based on Section 23 , pages 128 - 135 of the book Differential equations with applications and historical notes by Geroge F. Simmons, Second edition, Tata McGraw-Hill, New Delhi.