VECTOR CALCULUS, ANALYTIC GEOMETRY AND ABSTRACT ALGEBRA

Dr. Manju K Menon
Mathematics
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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 1-4  
Studies arc length and do some problems

 

3 Sessions 5-7  
Studies  Curvature and do some problems 3 Sessions 8-10  
Studies  Normal and Osculating Circle and do some problems

 

3 Sessions 11-13  
Understands Directional Derivative 5 Sessions 14-18  
Seminars, Additional Problems, Revision 3 Sessions 19-21  
Unit Test 1 Session 22  
 

Module II

 

Line Integrals 2 Sessions 23-24  
Work, Circulation and Flux

 

5 Sessions 25-29  
Conservation Fields and Potential Functions 3 Sessions 30-32  
Green’s theorem in Plane 4 Sessions 33-36  
Surface area and Surface integral 3 Sessions 37-39  
Stoke’s theorem 3 Sessions 40-42  
Divergence theorem 3 Sessions 43-45  
Additional Problems, Seminars, Unit Test 2 Sessions 46-47  
 

Module III

 

Understands polar coordinates 4

 

 

Sessions 48-51  
Studies different types of Conic Sections and their properties 10 Sessions 52-61  
Additional Problems, Seminars, Unit Test 4 Sessions 62-65  
 

Module IV

 

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

 

 

 

 

 

Sessions 66-69  
Understands the concept of subgraphs and do some problems 3 Sessions 70-72  
Understands permutations in graphs and see some examples

 

5 Sessions 73-77  
Understands the concept homomorphism and extend the study to isomorphism of groups 4 Sessions 78-81  
Additional Problems, Seminars, Unit Test 3 Sessions 82-84  

 

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 :

  1. George B. Thomas, Jr: Thomas’ Calculus Twelfth Edition, Pearson.

 

  1. John B Fraleigh – A First course in Abstract Algebra (Seventh Edition)

 

 REFERENCE BOOKS:

  1. Harry F. Davis & Arthur David Snider: Introduction to Vector Analysis, 6th ed.,

Universal Book Stall, New Delhi.

  1. Murray R. Spiegel: Vector Analysis, Schaum’s Outline Series, Asian Student edition.
  2. I.N. Herstein – Topics in Algebra
  3. Joseph A Gallian – A Contemporary Abstract Algebra, Narosa Publishing House.

WEB REFERENCE:

 

 

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.

    No items in this section
  • 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

    No items in this section
  • Module III: Analytic Geometry

    Polar coordinates, Conic sections, Conics in Polar coordinates.

    No items in this section
  • Module IV: Abstract algebra

    Groups, Subgroups, Cyclic groups, Groups of Permutations, Homomorphism.

    No items in this section
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Dr. Manju K Menon
Dr. Manju K. Menon joined St. Paul's College, Kalamassery in 2005. She was an NBHM Research Fellow in 2002-2005 and completed her Ph D in Graph Theory under the guidance of Prof. A. Vijayakumar, CUSAT in 2010. She has 10 publications to her credit and successfully completed a Minor Research Project Funded by UGC. She was the convener of two UGC Sponsored National Seminars and a Compact Course organised in association with the Ramanujan Mathematical Society. She is currently the Controller of Examinations of St. Paul's College, Kalamassery. She is a life member of RMS, IMS and KMA. She is also a resource person in many National and International Seminars including the Third Indo Taiwan International Conference on Discrete Mathematics held at Taiwan in 2013. She is a research guide in Mathematics in Maharajas College, Ernakulam affiliated to Mahathma Gandhi University, Kottayam.

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₹1,000.00