Syllabus (MATHEMATICS)

Course Type: MAJ-4

Semester: 4

Course Code: BMTMMAJ04T

Course Title: Abstract Algebra-I & Multivariate Calculus

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: TO BE DONE

Learning Outcome: The whole course will have the following outcomes: Upon successful completion of this course, the students will be able to CO1: acquire knowledge of group such as permutation group, symmetric group, quaternion group etc. CO2: understand Lagrange’s, Ferm

Syllabus:

Unit -1: Abstract Algebra-I [Credit: 3, 45L]

• Subgroup – Necessary and sufficient condition. Subgroup generated by a subset. Finite Group. Simple examples. Centralizer, Normalizer, Center of a group, Product of two subgroups.
• Symmetries of a square, Dihedral groups, Permutation groups, Symmetric groups and Quaternion groups (through matrices).
• Cyclic group, properties of cyclic groups, classification of subgroups of cyclic groups, Cycle notation for permutations, properties of permutations, even and odd permutations, Alternating group, properties of cosets, Lagrange’s theorem and consequences including Fermat’s Little theorem.
• Normal Subgroups, Quotient Group.
• Group homomorphism and isomorphism and their properties. Cayley’s theorem. First, Second and Third Isomorphism Theorems. Correspondence theorem for groups.
• Subrings, Ideals, Divisor of zero, Unit, Idempotent element, Nilpotent element, Integral Domains, Quotient Ring, Quotient field, Characteristic of a Ring, Subfield. Ring Homomorphism, Isomorphism. First, Second and Third Isomorphism theorems.

Unit -2: Multivariate Calculus [Credit: 3, 45L]

• Functions of several variables, limit and continuity of functions of two or more variables.
• Partial differentiation, total differentiability, sufficient condition for differentiability, Schwartz and Young’s theorems (statement only) and their verification. Directional derivatives, the gradient, Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems.
• Double integration over rectangular region, double integration over non-rectangular region, Double integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelepiped and solid regions. Volume by triple integrals, cylindrical and spherical co-ordinates. Change of order in double integrals and triple integrals.

Reading References:

1. Sen, Ghosh, Mukhopadhaya, Maity; Topics in Abstract Algebra; Universities Press.
2. S. K. Mapa, Higher Algebra (Abstract and Linear), Levant.
3. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
4. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
5. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., 1999, Narosa Publishing House, New Delhi.
6. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.
7. D.S. Malik, John M. Mordeson and M.K. Sen, Fundamentals of abstract algebra, McGraw-Hill Education-Europe.
8. Tom M. Apostol, Mathematical Analysis, Narosa Book Distributors Pvt. Ltd.-New Delhi.
9. Shanti Narayan, P.K. Mittal, Integral Calculus, S. Chand.
10. Maity & Ghosh, Integral Calculus, New Central Book Agency (P) Ltd.