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VECTOR CALCULUS, ANALYTIC GEOMETRY AND ABSTRACT ALGEBRA
Course Code: MM3CMT01 Coordinator: Dr. Manju K. Menon
Course Credit: 4
Semester: III
Class: BSc Physics and BSc Chemistry
Course Type: Complimentary
Department: Mathematics
COURSE OBJECTIVES
To understand the vector differentiation, Integration, Basics of Analytic Geometry and some basic Concepts in Abstract Algebra giving importance to do more problems.
Learning Outcomes
After completing this course the learner should be able to
 Analyze vector functions to find derivatives, tangent lines, integrals, arc length, and curvature
 Compute limits and derivatives of functions of 2 and 3 variables
 Apply derivative concepts to find tangent lines to level curves and solve optimization problems
 Evaluate double and triple integrals for area and volume
 Differentiate vector fields.
 Determine gradient vector fields and find potential functions
 Analyse the fundamental theorem of calculus and see their relation to fundamental theorem of Calculus , leading to the more generalised version of Stokes theorem in the setting of differential forms.
 Evaluate line integrals directly and by the fundamental theorem.
 Understand analytic Geometry
 Understands basics of Abstract Algebra
Syllubus
Module I: Vector valued Functions (15 hrs)
Curves in space and their tangents, Arc length in space, Curvature and Normal Vectors of a
curve, Directional Derivatives and Gradient Vectors.
Text 1: Chapter 13 (Sections 13.1, 13.3 and 13.4), Chapter 14 (Section 14.5 only)
Module II: Integration in Vector Fields (25hrs)
Line Integrals, Vector fields and line integrals: Work, Circulation and Flux. Path independence,
Conservation Fields and Potential Functions , Green’s theorem in Plane (Statement and problems only),
Surface area and Surface integral, Stoke’s theorem( Statement and Problems only), the Divergence
theorem and a Unified theory ( Statement and simple problems only).
Text 1: Chapter 16 (Sections 16.1 to 16.8)
Module III: Analytic Geometry (25 hrs)
Polar coordinates, Conic sections, Conics in Polar coordinates.
Text 1: Chapter 11 (Sections 11.3, 11.6 and 11.7)
Module IV: Abstract algebra (25 hrs)
Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism.
Text 2: Chapter 1 Sections 4, 5 and 6 (Proofs of Theorems/ Corollary 5.17, 6.3, 6.7, 6.10, 6.14, 6.16 are excluded)
Chapter 2, Section 8 (Proofs of theorems 8.15 and 8.16 are excluded)
COURSE OUTLINE
Module
No 
Topics  Learning Outcomes  Hours 
1  Curves in space and their tangents
Arc length in space, Curvature and Normal Vectors of acurve, Directional Derivatives and Gradient Vectors.

· Studies different curves and their graphs. Also studies the tangent equations
· Studies arc length and do some problems · Studies Curvature and do some problems · Studies Normal and Osculating Circle and do some problems · Understands Directional Derivative 
4
3
3
3
5 
Seminar will be given to ¼ th of students  
2  Line Integrals, Vector fields and line integrals: Work, Circulation and Flux. Path independence,
Conservation Fields and Potential Functions , Green’s theorem in Plane (Statement and problems only), Surface area and Surface integral, Stoke’s theorem( Statement and Problems only), the Divergence theorem and a Unified theory ( Statement and simple problems only).

· Line Integrals
· Work, Circulation and Flux · Conservation Fields and Potential Functions · Green’s theorem in Plane · Surface area and Surface integral · Stoke’s theorem · Divergence theorem 
2
5
3
4
3
3 3 
Seminar will be given to ¼ th of students  
Internal I  
3  Polar coordinates, Conic sections, Conics in Polar coordinates.  · Understands polar coordinates
· Studies different types of Conic Sections and their properties 
4
10 
Seminar will be given to ¼ th of students students  
4  Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism  · Studies Groups, Definition and its basic properties. Sees some examples.
· Understands the concept of subgraphs and do some problems · Understands permutations in graphs and see some examples · Understands the concept homomorphism and extend the study to isomorphism of groups 
4
3
5
4 
Seminar will be given to ¼ th of students students  
Internal II 
TEACHING SCHEDULE
TOPICS  HOURS  SESSIONS  DESCRIPTION  
Module I


Studies different curves and their graphs. Also studies the tangent equations 
4

Sessions 14  
Studies arc length and do some problems

3  Sessions 57  
Studies Curvature and do some problems  3  Sessions 810  
Studies Normal and Osculating Circle and do some problems

3  Sessions 1113  
Understands Directional Derivative  5  Sessions 1418  
Seminars, Additional Problems, Revision  3  Sessions 1921  
Unit Test  1  Session 22  
Module II


Line Integrals  2  Sessions 2324  
Work, Circulation and Flux

5  Sessions 2529  
Conservation Fields and Potential Functions  3  Sessions 3032  
Green’s theorem in Plane  4  Sessions 3336  
Surface area and Surface integral  3  Sessions 3739  
Stoke’s theorem  3  Sessions 4042  
Divergence theorem  3  Sessions 4345  
Additional Problems, Seminars, Unit Test  2  Sessions 4647  
Module III


Understands polar coordinates  4

Sessions 4851  
Studies different types of Conic Sections and their properties  10  Sessions 5261  
Additional Problems, Seminars, Unit Test  4  Sessions 6265  
Module IV


Studies Groups, Definition and its basic properties. Sees some examples.  4

Sessions 6669  
Understands the concept of subgraphs and do some problems  3  Sessions 7072  
Understands permutations in graphs and see some examples

5  Sessions 7377  
Understands the concept homomorphism and extend the study to isomorphism of groups  4  Sessions 7881  
Additional Problems, Seminars, Unit Test  3  Sessions 8284  
PEDAGOGY
After the course the students will understand vector differentiation, double and triple integration, analytic geometry and abstract algebra. The tutorials will focus on the understanding of concepts by doing moreproblems.
EVALUATION STRATEGY
The internal evaluation is based on the students attendance, seminar, Internal Assessments and assignment marks.
EVALUATION SCHEME
Sl. No  Components  Weightage 
1  END TERM EXAM  80 
2  ATTENDANCE  5 
3  INTERNAL MARKS  10 
4  SEMINARS/ ASSIGNMENT/ VIVA  5 
TEXT BOOK :
 George B. Thomas, Jr: Thomas’ Calculus Twelfth Edition, Pearson.
 John B Fraleigh – A First course in Abstract Algebra (Seventh Edition)
REFERENCE BOOKS:
 Harry F. Davis & Arthur David Snider: Introduction to Vector Analysis, 6th ed.,
Universal Book Stall, New Delhi.
 Murray R. Spiegel: Vector Analysis, Schaum’s Outline Series, Asian Student edition.
 I.N. Herstein – Topics in Algebra
 Joseph A Gallian – A Contemporary Abstract Algebra, Narosa Publishing House.
WEB REFERENCE:
 https://www.youtube.com/watch?v=kdTxN4E0vbo
 https://www.youtube.com/watch?v=QSzUSunxA
 https://www.youtube.com/watch?v=eSqznPrtzS4
 https://www.youtube.com/watch?v=kQPeyobn0UM
 https://www.youtube.com/watch?v=Z9XL4B0ROk
FACULTY DETAILS
Dr. Manju K. Menon
Website: stpauls.ac.in
Email: [email protected]
Mobile: 9846335837
SEMINARS/ASSIGNMENTS
Topics will be announced in the online/offline class during the course.

Module I: Vector valued Functions
Curves in space and their tangents, Arc length in space, Curvature and Normal Vectors of a curve, Directional Derivatives and Gradient Vectors.

Module II: Integration in Vector Fields
Line Integrals, Vector fields and line integrals: Work, Circulation and Flux. Path independence, Conservation Fields and Potential Functions , Green’s theorem in Plane (Statement and problems only), Surface area and Surface integral, Stoke’s theorem( Statement and Problems only), the Divergence theorem and a Unified theory

Module III: Analytic Geometry
Polar coordinates, Conic sections, Conics in Polar coordinates.

Module IV: Abstract algebra
Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism.
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