Semester 2: Syllabus B.Sc. in Mathematics Honours (CBCS): Vidyasagar University
Semester 2
Paper Code: C3T (Credit: 6, Full Marks: 75)
Paper Name: Real Analysis
Unit 1
- Review of algebraic and order properties of \( \mathbb{R} \), ε-neighborhood of a point in \( \mathbb{R} \). Idea of countable sets, uncountable sets and uncountability of \( \mathbb{R} \). Bounded above sets, bounded below sets, bounded sets, unbounded sets. Suprema and infima. Completeness property of \( \mathbb{R} \) and its equivalent properties. The Archimedean property, density of rational (and irrational) numbers in \( \mathbb{R} \), intervals. Limit points of a set, isolated points, open set, closed set, derived set, illustrations of Bolzano-Weierstrass theorem for sets, compact sets in \( \mathbb{R} \), Heine-Borel Theorem.
Unit 2
- Sequences, bounded sequence, convergent sequence, limit of a sequence, liminf, lim sup. Limit theorems. Monotone sequences, monotone convergence theorem. Subsequences, divergence criteria. Monotone subsequence theorem (statement only), Bolzano-Weierstrass theorem for sequences. Cauchy sequence, Cauchy’s convergence criterion.
Unit 3
- Infinite series, convergence and divergence of infinite series, Cauchy criterion, tests for convergence: comparison test, limit comparison test, ratio test, Cauchy’s nth root test, integral test. Alternating series, Leibniz test. Absolute and conditional convergence.
Graphical Demonstration (Teaching Aid)
- Plotting of recursive sequences.
- Study the convergence of sequences through plotting.
- Verify Bolzano-Weierstrass theorem through plotting of sequences and hence identify convergent subsequences from the plot.
- Study the convergence/divergence of infinite series by plotting their sequences of partial sum.
- Cauchy’s root test by plotting nth roots.
- Ratio test by plotting the ratio of nth and (n+1)th term.
Suggested Books
- R.G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.
- Gerald G. Bilodeau, Paul R. Thie, G.E. Keough, An Introduction to Analysis, 2nd Ed., Jones & Bartlett, 2010.
- Brian S. Thomson, Andrew. M. Bruckner and Judith B. Bruckner, Elementary Real Analysis, Prentice Hall, 2001.
- S.K. Berberian, A First Course in Real Analysis, Springer Verlag, New York, 1994.
- T. Apostol, Mathematical Analysis, Narosa Publishing House.
- Courant and John, Introduction to Calculus and Analysis, Vol I, Springer.
- W. Rudin, Principles of Mathematical Analysis, Tata McGraw-Hill.
- Terence Tao, Analysis I, Hindustan Book Agency, 2006.
- S. Goldberg, Calculus and Mathematical Analysis.
Semester 2
Paper Code: C4T (Credit: 6, Full Marks: 75)
Paper Name: Differential Equations & Vector Calculus
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 non-homogeneous 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.
Graphical Demonstration (Teaching Aid)
- 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 problemsComputing 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 Mc Graw 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.
Semester 2
Paper Code: GE2T (Credit: 6, Full Marks: 75)
Paper Name: Algebra
Unit 1
- Polar representation of complex numbers, nth roots of unity, De Moivre’s theorem for rational indices and its applications.
- Theory of equations: Relation between roots and coefficients, transformation of equation, Descartes rule of signs, cubic and biquadratic equation.
- Inequality: The inequality involving AM≥ GM≥ HM, Cauchy-Schwartz inequality.
Unit 2
- Equivalence relations. Functions, composition of functions, Invertible functions, one to one correspondence and cardinality of a set.
- Well-ordering property of positive integers, division algorithm, divisibility and Euclidean algorithm. Congruence relation between integers. Principles of Mathematical induction, statement of Fundamental Theorem of Arithmetic.
Unit 3
- Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation Ax=b, solution sets of linear systems, applications of linear systems, linear independence.
Unit 4
- Introduction to linear transformations, matrix of a linear transformation, inverse of a matrix, characterizations of invertible matrices.
- Subspaces of Rn, dimension of subspaces of Rn, rank of a matrix, Eigen values, eigen vectors and characteristic equation of a matrix.
- Cayley-Hamilton theorem and its use in finding the inverse of a matrix.
Suggested Books
- Titu Andreescu and Dorin Andrica, Complex Numbers from A to Z, Birkhauser, 2006.
- Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005.
- David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian Reprint, 2007.
- K.B. Dutta, Matrix and linear algebra.
- K. Hoffman, R. Kunze, Linear algebra.
- W.S. Burnstine and A.W. Panton, Theory of equations.
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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.
