Semester 3: Syllabus B.Sc. in Mathematics Honours (CBCS): Vidyasagar University
Semester 3
Paper Code: C5T (Credit: 6, Full Marks: 75)
Paper Name: Theory of Real Functions & Introduction to Metric Space
Unit 1
- Limits of functions (ε – δ approach), sequential criterion for limits, divergence criteria. Limit theorems, one sided limits. Infinite limits and limits at infinity.
- Continuous functions, sequential criterion for continuity and discontinuity. Algebra of continuous functions. Continuous functions on an interval, intermediate value theorem, location of roots theorem, preservation of intervals theorem.
- Uniform continuity, non-uniform continuity criteria, uniform continuity theorem.
Unit 2
- Differentiability of a function at a point and in an interval, Caratheodory’s theorem, algebra of differentiable functions.
- Relative extrema, interior extremum theorem. Rolle’s theorem. Mean value theorem, intermediate value property of derivatives, Darboux’s theorem. Applications of mean value theorem to inequalities and approximation of polynomials.
Unit 3
- Cauchy’s mean value theorem. Taylor’s theorem with Lagrange’s form of remainder, Taylor’s theorem with Cauchy’s form of remainder, application of Taylor’s theorem to convex functions, relative extrema.
- Taylor’s series and Maclaurin’s series expansions of exponential and trigonometric functions, ln (1 + x), 1/(ax + b) and (x + 1)n. Application of Taylor’s theorem to inequalities.
Unit 4
- Metric spaces: Definition and examples. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, subspaces, dense sets, separable spaces.
Suggested Books
- R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003.
- K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2004.
- A. Mattuck, Introduction to Analysis, Prentice Hall, 1999.
- S.R. Ghorpade and B.V. Limaye, a Course in Calculus and Real Analysis, Springer, 2006.
- T. Apostol, Mathematical Analysis, Narosa Publishing House.
- Courant and John, Introduction to Calculus and Analysis, Vol II, Springer.
- W. Rudin, Principles of Mathematical Analysis, Tata McGraw-Hill.
- Terence Tao, Analysis II, Hindustan Book Agency, 2006.
- Satish Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London, 2006.
- S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing House, 2011.
- G.F. Simmons, Introduction to Topology and Modern Analysis, McGrawHill, 2004.
Semester 3
Paper Code: C6T (Credit: 6, Full Marks: 75)
Paper Name: Group Theory 1
Unit 1
- Symmetries of a square, dihedral groups, definition and examples of groups including permutation groups and quaternion groups (through matrices), elementary properties of groups.
Unit 2
- Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups.
Unit 3
- 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.
Unit 4
- External direct product of a finite number of groups, normal subgroups, factor groups, Cauchy’s theorem for finite abelian groups.
Unit 5
- Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of isomorphisms. First, Second and Third isomorphism theorems.
Suggested Books
- John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
- M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
- Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.
- Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995.
- I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.
- D.S. Malik, John M. Mordeson and M.K. Sen, Fundamentals of abstract algebra.
Semester 3
Paper Code: C7T (Credits: 04)
Paper Name: Numerical Methods
Unit 1
- Algorithms. Convergence. Errors: relative, absolute. Round off. Truncation.
Unit 2
- Transcendental and polynomial equations: Bisection method, Newton’s method, secant method, Regula-falsi method, fixed point iteration, Newton-Raphson method. Rate of convergence of these methods.
Unit 3
- System of linear algebraic equations: Gaussian elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis. LU decomposition.
Unit 4
- Interpolation: Lagrange and Newton’s methods. Error bounds. Finite difference operators. Gregory forward and backward difference interpolation.
- Numerical differentiation: Methods based on interpolations, methods based on finite differences.
Unit 5
- Numerical Integration: Newton Cotes formula, Trapezoidal rule, Simpson’s 1/3rd rule, Simpsons 3/8th rule, Weddle’s rule, Boole’s Rule. midpoint rule, Composite trapezoidal rule, composite Simpson’s 1/3rd rule, Gauss quadrature formula.
- The algebraic eigen value problem: Power method.
- Approximation: Least square polynomial approximation.
Unit 6
- Ordinary differential equations: The method of successive approximations, Euler’s method, the modified Euler method, Runge-Kutta methods of orders two and four.
Suggested Books
- Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007.
- M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 6th Ed., New age International Publisher, India, 2007.
- C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education, India, 2008.
- Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, 7th Ed., PHI Learning Private Limited, 2013.
- John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012.
Paper Code: C7P (Credits: 02)
Paper Name: Numerical Methods Laboratory
Lab 1
- Implementation of various numerical methods using programming languages like Python, MATLAB, or C.
- Exercises on solving equations, system of linear equations, interpolation, and numerical integration.
- Practical implementation of error analysis and convergence tests.
Lab 2
- Using software tools for numerical analysis and solving practical problems.
- Application of numerical methods to real-world problems and case studies.
Suggested Books
- John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012.
- Steven C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw-Hill, 2009.
- Ralph M. Parker, Numerical Analysis, McGraw-Hill, 2008.
Semester 3
Paper Code: SEC1T (Credits: 02)
Paper Name: Object Oriented Programming in C++
Unit 1
- Programming paradigms, characteristics of object oriented programming languages, brief history of C++, structure of C++ program, differences between C and C++, basic C++ operators, Comments, working with variables, enumeration, arrays and pointer.
Unit 2
- Objects, classes, constructor and destructors, friend function, inline function, encapsulation, data abstraction, inheritance, polymorphism, dynamic binding, operator overloading, method overloading, overloading arithmetic operator and comparison operators.
Unit 3
- Template class in C++, copy constructor, subscript and function call operator, concept of namespace and exception handling.
Suggested Books
- A. R. Venugopal, Rajkumar, and T. Ravishanker, Mastering C++, TMH, 1997.
- S. B. Lippman and J. Lajoie, C++ Primer, 3rd Ed., Addison Wesley, 2000.
- Bruce Eckel, Thinking in C++, 2nd Ed., President, Mindview Inc., Prentice Hall.
- D. Parasons, Object Oriented Programming with C++, BPB Publication.
- Bjarne Stroustrup, The C++ Programming Language, 3rd Ed., Addison Welsley.
- E. Balaguruswami, Object Oriented Programming In C++, Tata McGrawHill.
- Herbert Schildt, C++, The Complete Reference, Tata McGrawHill.
Paper Code: SEC1T (Credits: 02)
Paper Name: Logic and Sets
Unit 1
- Introduction, propositions, truth table, negation, conjunction and disjunction. Implications, biconditional propositions, converse, contra positive and inverse propositions and precedence of logical operators. Propositional equivalence: Logical equivalences. Predicates and quantifiers: Introduction, quantifiers, binding variables and negations.
Unit 2
- Sets, subsets, set operations and the laws of set theory and Venn diagrams. Examples of finite and infinite sets. Finite sets and counting principle. Empty set, properties of empty set. Standard set operations. classes of sets. Power set of a set.
Unit 3
- Difference and Symmetric difference of two sets. Set identities, generalized union and intersections. Relation: Product set. Composition of relations, types of relations, partitions, equivalence Relations with example of congruence modulo relation. Partial ordering relations, n- ary relations.
Suggested Books
- R.P. Grimaldi, Discrete Mathematics and Combinatorial Mathematics, Pearson Education, 1998.
- P.R. Halmos, Naive Set Theory, Springer, 1974.
- E. Kamke, Theory of Sets, Dover Publishers, 1950.
Paper Code: GE3T – Differential Equations & Vector Calculus (Credits: 06)
Paper Name: Differential Equations & Vector Calculus
Differential Equations
Unit 1
- Lipschitz condition and Picard’s Theorem (Statement only). General solution of homogeneous equation of second order, principle of superposition for homogeneous equation, Wronskian: its properties and applications, Linear homogeneous and nonhomogeneous equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters.
Unit 2
- Systems of linear differential equations, types of linear systems, differential operators, an operator method for linear systems with constant coefficients, Basic Theory of linear systems in normal form, homogeneous linear systems with constant coefficients: Two Equations in two unknown functions.
Unit 3
- Equilibrium points, Interpretation of the phase plane, Power series solution of a differential equation about an ordinary point, solution about a regular singular point.
Unit 4
- Triple product, introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, differentiation and integration of vector functions.
Unit 5
- Plotting of family of curves which are solutions of second order differential equation.
- Plotting of family of curves which are solutions of third order differential equation.
Suggested Books
- Belinda Barnes and Glenn R. Fulford, Mathematical Modeling with Case Studies, A Differential Equation Approach using Maple and Matlab, 2nd Ed., Taylor and Francis group, London and New York, 2009.
- C.H. Edwards and D.E. Penny, Differential Equations and Boundary Value problems Computing and Modeling, Pearson Education India, 2005.
- S.L. Ross, Differential Equations, 3rd Ed., John Wiley and Sons, India, 2004.
- Martha L Abell, James P Braselton, Differential Equations with MATHEMATICA, 3rd Ed., Elsevier Academic Press, 2004.
- Murray, D., Introductory Course in Differential Equations, Longmans Green and Co.
- Boyce and Diprima, Elementary Differential Equations and Boundary Value Problems, Wiley.
- G.F. Simmons, Differential Equations, Tata McGraw Hill.
- Marsden, J., and Tromba, Vector Calculus, McGraw Hill.
- Maity, K.C. and Ghosh, R.K. Vector Analysis, New Central Book Agency (P) Ltd. Kolkata (India).
- M.R. Speigel, Schaum’s outline of Vector Analysis.
Paper Code: GE3T – Group Theory 1 (Credits: 06)
Paper Name: Group Theory 1
Group Theory 1
Unit 1
- Symmetries of a square, dihedral groups, definition and examples of groups including permutation groups and quaternion groups (through matrices), elementary properties of groups.
Unit 2
- Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups.
Unit 3
- 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.
Unit 4
- External direct product of a finite number of groups, normal subgroups, factor groups, Cauchy’s theorem for finite abelian groups.
Unit 5
- Group homomorphisms, properties of homomorphisms, Cayley’s theorem, properties of isomorphisms. First, Second and Third isomorphism theorems.
Suggested Books
- John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
- M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
- Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.
- Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer.
Paper Code: GE3T – Theory of Real Functions & Introduction to Metric Space (Credits: 06)
Paper Name: Theory of Real Functions & Introduction to Metric Space
Theory of Real Functions & Introduction to Metric Space
Unit 1
- Limits of functions (ε – δ approach), sequential criterion for limits, divergence criteria. Limit theorems, one sided limits. Infinite limits and limits at infinity. Continuous functions, sequential criterion for continuity and discontinuity. Algebra of continuous functions. Continuous functions on an interval, intermediate value theorem, location of roots theorem, preservation of intervals theorem. Uniform continuity, non-uniform continuity criteria, uniform continuity theorem.
Unit 2
- Differentiability of a function at a point and in an interval, Caratheodory’s theorem, algebra of differentiable functions. Relative extrema, interior extremum theorem. Rolle’s theorem. Mean value theorem, intermediate value property of derivatives, Darboux’s theorem. Applications of mean value theorem to inequalities and approximation of polynomials.
Unit 3
- Cauchy’s mean value theorem. Taylor’s theorem with Lagrange’s form of remainder, Taylor’s theorem with Cauchy’s form of remainder, application of Taylor’s theorem to convex functions, relative extrema. Taylor’s series and Maclaurin’s series expansions of exponential and trigonometric functions, ln (1 + x), 1/(ax + b) and (x+1)n. Application of Taylor’s theorem to inequalities.
Unit 4
- Metric spaces: Definition and examples. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, subspaces, dense sets, separable spaces.
Suggested Books
- R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003.
- K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2004.
- A. Mattuck, Introduction to Analysis, Prentice Hall, 1999.
- S.R. Ghorpade and B.V. Limaye, A Course in Calculus and Real Analysis, Springer, 2006.
- Walter Rudin, Principles of Mathematical Analysis, McGraw Hill, 1976.
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Frequently Asked Questions (FAQs)
What is the Choice Based Credit System (CBCS)?
The Choice Based Credit System (CBCS) is an educational system that offers flexibility in the choice of courses for students. It allows students to choose from a range of courses and accumulate credits based on the courses they complete. This system promotes a more comprehensive learning experience and helps students tailor their education according to their interests and career goals.
What are the core courses for the B.Sc. Honours in Mathematics at Vidyasagar University?
The core courses for the B.Sc. Honours in Mathematics are:
- Calculus, Geometry & Differential Equation
- Algebra
- Real Analysis
- Differential Equations & Vector Calculus
- Theory of Real Functions & Introduction to Metric Space
- Group Theory I
- Numerical Methods
- Riemann Integration and Series of Functions
- Multivariate Calculus
- Ring Theory and Linear Algebra I
- Partial Differential Equations & Applications
- Group Theory II
- Metric Spaces and Complex Analysis
- Ring Theory and Linear Algebra II
What are the Discipline Specific Electives (DSE) available?
The Discipline Specific Electives (DSE) for the B.Sc. Honours in Mathematics include:
- Linear Programming or Point Set Topology or Theory of Equations
- Probability & Statistics or Boolean Algebra and Automata Theory or Portfolio Optimization
- Mechanics or Number Theory or Industrial Mathematics
- Mathematics Modeling or Differential Geometry or Bio Mathematics
What are the Skill Enhancement Courses (SEC) offered?
The Skill Enhancement Courses (SEC) include:
- Object Oriented Programming in C++ or Logic & Sets
- Graph Theory or Computer Graphics or Operating System: Linux
What is the structure of the B.Sc. Honours Mathematics program?
The B.Sc. Honours Mathematics program is structured across six semesters with Core Courses, Discipline Specific Electives (DSE), Skill Enhancement Courses (SEC), and Ability Enhancement Compulsory Courses (AECC). Each semester has a mix of theoretical and practical components, and students are evaluated through Continuous Assessment (CA) and End Semester Examination (ESE).
How many total credits are required to complete the B.Sc. Honours in Mathematics?
The total number of credits required to complete the B.Sc. Honours in Mathematics is 142 credits across all six semesters.
Can I choose courses from other disciplines?
Yes, as part of the Generic Electives (GE), you can choose courses from other disciplines based on availability and departmental regulations.
What is the role of Ability Enhancement Compulsory Courses (AECC)?
Ability Enhancement Compulsory Courses (AECC) are designed to improve students’ skills and knowledge in areas such as English and Environmental Studies. These courses are mandatory and contribute to the overall credit requirement for the degree.
How are the courses evaluated in the B.Sc. Honours Mathematics program?
Courses are evaluated through a combination of Continuous Assessment (CA) and End Semester Examination (ESE). CA includes quizzes, assignments, and projects, while ESE consists of final exams at the end of each semester.
What is the duration of the B.Sc. Honours in Mathematics program?
The B.Sc. Honours in Mathematics program is designed to be completed over three years, divided into six semesters.
Can I pursue a minor or additional specialization in the program?
The program primarily focuses on Mathematics as a major. However, students can explore additional specializations or minors depending on the availability of courses and university regulations.
Are there any research opportunities in the B.Sc. Honours program?
Research opportunities are generally available through projects and assignments in higher semesters. Students interested in research should consult with their professors or academic advisors for specific opportunities.
What are the career prospects after completing the B.Sc. Honours in Mathematics?
Graduates can pursue careers in various fields such as education, finance, data analysis, actuarial science, and more. Further studies, such as a Master’s degree or professional certifications, can also enhance career prospects.
How can I access the course materials and syllabus?
Course materials and syllabus can typically be accessed through the university’s online portal or by contacting the respective department. Ensure you are enrolled and have the necessary credentials for access.
Are there any mandatory internships or projects?
While not always mandatory, internships and project work may be encouraged or required depending on specific courses or departmental guidelines. Check the course structure and departmental requirements for detailed information.
What are the benefits of studying Mathematics at the Honours level?
Studying Mathematics at the Honours level provides a deep understanding of mathematical concepts, problem-solving skills, and analytical abilities. It prepares students for advanced studies or careers in various fields where mathematics is applied.
How can I contact academic advisors or professors?
You can contact academic advisors or professors through the university’s official communication channels, such as email or the academic department office. Make sure to follow any specified procedures for scheduling meetings or consultations.
What resources are available for students in the Mathematics program?
Resources available include textbooks, online journals, library access, research papers, and departmental seminars. Students can also benefit from study groups, tutoring services, and online educational platforms.
Are there any scholarships available for Mathematics students?
Scholarships may be available based on academic performance, financial need, or other criteria. Check with the university’s financial aid office or the Mathematics department for information on available scholarships and application procedures.
What are the prerequisites for enrolling in the B.Sc. Honours in Mathematics?
Prerequisites typically include a strong background in Mathematics from secondary education. Specific requirements may vary, so it’s best to check the admission criteria outlined by the university.
How do I apply for the B.Sc. Honours in Mathematics program?
Application procedures can be found on the university’s admissions website. Follow the outlined steps, including filling out the application form, providing necessary documents, and meeting application deadlines.
Is there a placement cell for assisting with job placements?
Many universities have a placement cell that helps students with job placements and internships. Contact the university’s career services or placement office for information on available support and resources.
What is the maximum number of credits I can take in a semester?
The maximum number of credits per semester is usually defined by the university’s academic regulations. Check the specific guidelines provided by the university or consult with an academic advisor for detailed information.
Can I switch to another major or program after enrolling?
Switching majors or programs may be possible depending on university policies and availability. Consult with the academic advisor or registrar’s office to understand the procedures and requirements for changing your program.
How can I improve my chances of academic success in this program?
To improve academic success, focus on attending classes regularly, participating actively, managing your time effectively, and seeking help from professors or peers when needed. Utilize available resources and stay organized with your study materials.
Are there any extracurricular activities related to Mathematics?
Extracurricular activities may include math clubs, competitions, seminars, and workshops. Check with the Mathematics department or student organizations for opportunities to participate in activities related to your field of study.
What should I do if I need academic assistance?
If you need academic assistance, seek help from your professors, academic advisors, or tutoring services offered by the university. Additionally, participating in study groups and utilizing online resources can be beneficial.
How is the B.Sc. Honours in Mathematics different from a general B.Sc. in Mathematics?
The B.Sc. Honours program typically involves a more in-depth study of Mathematics with a focus on core and elective courses. It often requires a higher number of credits and includes more specialized courses compared to a general B.Sc. program.
What is the policy on attendance for this program?
Attendance policies are set by the university and may vary by course. Generally, regular attendance is required to meet academic standards and avoid penalties. Check the specific attendance policy for each course in the program.
How can I provide feedback about the courses or program?
Feedback can typically be provided through course evaluations, student surveys, or directly to the department or academic advisors. Your feedback helps improve the quality of the program and the educational experience.
