# TEW - Algebra & Multivariate Calculus (2021)

## Speakers and Syllabus

Syllabus to be covered in terms of modules of 6 lectures each.

 Name of the Speaker with affiliation, who will covereach module of 6 lectures. No. of Lectures Detailed Syllabus Prof. C.S. Dalawat, Harish Chandra Research Institute, Prayagraj 6 Vector spaces and their basic properties, Linear Transformations and Matrices, Change of Basis, Quotient Spaces, Rank and Nullity theorem. Determinants, Eigenvalues and Eigenvectors, Cayley- Hamilton Theorem, Decomposition theorems, Diagonalisation, Some application. Modules, Modules versus Vector Spaces, Free Modules, Matrices over PIDs, Smith Normal Form, Finitely Generated Modules over PIDs and their structure, Rational and Jordan Canonical Forms. Dr. Sanjay Kumar Singh, Indian Institute of Science Education and Research (IISER) Bhopal. 6 Symmetries of equilateral triangle and square; translations, rotations and reflections in the Euclidean plane. Group, subgroup, cyclic group, examples, Group homomorphism and normal subgroup, kernel and image of a homomorphism, quotient group, isomorphism theorems, Permutations of a finite set, permutation group Sn, Group action, class equation, Sylow’s Theorems and applications. Example of Matrix group over real and complex numbers. Group of symmetries of geometric objects in Euclidean spaces, dihedral group as the group of symmetries of a regular polygon, isometries of the Euclidean space, symmetries, platonic solids and their dual, symmetries of a tetrahedron, symmetries of a cube and octahedron, symmetries of icosahedron and dodecahedron, finite subgroups of SO(2) and SO(3). Dr. Siddhartha Sarkar, Indian Institute of Science Education and Research (IISER) Bhopal. 6 Basics of metric spaces, norms induced by inner products, Cauchy Schwarz inequality, equivalence of norms on finite dimensional vector spaces. Limits and continuity, Derivative as a linear map, examples from first principle, chain rule, directional derivative and partial derivative, examples. Sufficient condition for existence of total derivative, equality of mixed partial derivatives, Taylor’s theorem up to second order and applications to maxima and minima, method of Lagrange multipliers, Mean value and Implicit function theorem.

References:

1. Online Notes: http://www.mathe2.uni-bayreuth.de/stoll/lecture-notes/LinearAlgebraI.pdf
2. T.S. Blyth, Module Theory: An Approach To Linear Algebra
3. Mark A. Armstrong, Groups and Symmetry, Springer, 1997.
4. Online Notes on Groups and symmetry: http://www.maths.gla.ac.uk/~ajb/dvi-ps/2q-notes.pdf
5. Tom Apostol, Mathematical Analysis.
6. Walter Rudin, Principles of Mathematical Analysis.

Names of the Tutors / Course Associates with their affiliation and status:

1. Dr. Punam Gupta, (Assistant Professor)
Dr. Harisingh Gour Vishwavidyalaya.
2. Dr. Triloki Nath, (Assistant Professor)
Dr. Harisingh Gour Vishwavidyalaya.
3. Dr. R.K. Pandey, (Assistant Professor)
Dr. Harisingh Gour Vishwavidyalaya.

## Time Table

Tentative Time-Table:

 Day Date Lecture /Tutorial 5 PM- 6 PM Lecture /Tutorial 3.30 PM- 4.30 PM Friday 08/01/2021 L1: CSD Saturday 09/01/2021 L2: CSD Sunday 10/01/2021 L3: CSD T1:  PG,TN Friday 15/01/2021 L4: CSD Saturday 16/01/2021 L5: CSD Sunday 17/01/2021 L6: CSD T2:  PG,TN Friday 22/01/2021 L7: SKS Saturday 23/01/2021 L8: SKS Sunday 24/01/2021 L9: SKS T3: TN,RKP Friday 29/01/2021 L10: SS Saturday 30/01/2021 L11: SS Sunday 31/01/2021 L12: SS T4: TN,RKP Friday 05/02/2021 L13: SKS Saturday 06/02/2021 L14: SKS Sunday 07/02/2021 L15: SKS T5: PG,RKP Friday 12/02/2021 L16: SS Saturday 13/02/2021 L17: SS Sunday 14/02/2021 L18: SS T6: PG,RKP

CSD: Prof. C.S. Dalawat
SKS: Dr. Sanjay Kumar Singh
SS: Dr. Siddhartha Sarkar
PG: Dr. Punam Gupta
TN: Dr. Triloki Nath
RKP: Dr. R.K. Pandey

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