Class timings: TuTh 10:30-11:45AM
Venue: Old Physics Building, Room F-12
Tutorial: Fri 11:00-11:50AM (see below for sections)
Prerequisite: UMA 101
Book: Tom M. Apostol, Calculus, Volume 2, 2nd edition, Wiley, India Edition, 2001
MS Teams Code: l7s4ie6 (every registered student must join)
Grading policy: 30% Quizzes , 30% Midterm exam, 40% Final exam.
Quizzes will be based on the weekly homework, and will be held during the tutorials.
The best 10 quiz scores will be counted towards the final grade. There will be no make-up quizzes.
The usual rules for UG students regarding attendance, taking both midterm and final exam, and plagiarism will apply.
Section A
TA: Manpreet Singh Location: F-01 Office hours: Mon 5:30-6:30 at N-11
Section B
TA: Srikanth Pai Location: F-8A Office hours: Thu 5-6 at N-11
Section C
TA: S P Murugan Location: F-12 Office hours: Tue 6-7 at R-23
Section D
TA: Debaprasanna Kar Location: G-02 Office hours: Wed 6-7 at R-21
Section E
TA: Sumit Arora Location: G-01 Office hours: Fri 5:30-6:30 at N-28
Reserve TA: Vijaya Kumar U
Note: The TA offices are in the Mathematics department
The material covered weekly will be updated below as the semester progresses, and lecture notes will be available on Teams. The homework problem set for each week will be posted here, and will be announced on Teams. They are not meant for submission, but are absolutely essential to the course.
Week 1
Review of vector spaces (1.2 - 1.8), linear maps (2.1,2.2), Rank-Nullity Theorem (2.3). Matrices (2.10), solving linear equations (2.18).
Week 2
Gauss-Jordan elimination (2.18), Inner products and orthogonality (1.11-12), Gram-Schmidt orthogonalization (1.14).
Week 3
Orthogonal decomposition (1.15) , Approximation theorem (1.16), Least squares method (Lecture notes), Row rank = Column rank (Lecture notes), Determinants: axiomatic definition and some properties (3.1-4).
Week 4
Determinants: uniqueness, product formula, linear independence (3.5,3.7,3.9), existence: expansion formula along first column (3.13), expansion formula along first column, and determinant of transpose (3.12,3.14, supplementary notes) Eigenvalues and eigenvectors: definition and examples (4.1,4.2). Characteristic polynomial (4.5).
Week 5
Characteristic polynomial (continued) (4.5), examples of computing eigenvalues and eigenvectors (4.6,4.7), similar matrices (4.9). Symmetric and Hermitian matrices (5.1,5.2,5.8).
Week 6
Proof of Spectral Theorem for Hermitian matrices. Orthogonal and unitary matrices.
Week 7
Linear systems of differential equations, Review, preview of post-midterm material.
Midterm week
Week 8
Functions of several variables; examples, definition of an open set, limits and continuity, some examples. (Lecture notes - Lec 15) Properties of limits, Various examples (8.4, 8.10) , Directional derivatives (8.6,8.7), Definition of total derivative (8.11, to be continued).
Week 9
Total derivative: Jacobian matrix (8.11,8.12, Lecture notes), basic properties (8.19, Lecture notes) , chain rule: statement and applications (8.15.8.21), mean value theorem in several variables (Lecture notes), sufficient criterion for differentiability (8.13).
Week 10
Minima and Maxima for multivariable functions (9.9, Lecture notes), the Hessian and the second derivative test (9.10,9.11), Lagrange multiplier method (9.14), Line Integrals (10.2-4).
Week 11
Gradient vector fields and their line integrals (10.11, 10.16). Double integrals: definition (11.1-5). Integrability (11.10-11), Fubini's theorem (11.6). Examples (Lecture notes).
Week 12
Computing double integrals over regions (11.12), Examples/Applications (11.14), Green's theorem (11.19), Applications (11.20), Change of variables formula, examples (11.26, 11.27).
Week 13
More about the change of variables formula (Lecture notes), Triple integrals (11.31), Parametric surfaces, the fundamental vector product (12.2, 12.3) , area of a parametric surface, surface integrals, flux of a vector field across a surface (Lecture notes), statement of Stokes' theorem for surfaces.
Week 14
More on Stokes' theorem: curl of a vector field, compatible orientation, examples, special case is Green's theorem. Divergence theorem. (Lecture notes)