Syllabus for F.Y.BSc / B.A

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1 JAI HIND COLLEGE AUTONOMOUS Syllabus for F.Y.BSc / B.A Course : Mathematics Semester : I Credit Based Semester & Grading System With effect from Academic Year

2 List of Courses Course: FYBSc / BA Semester: I SR. NO. COURSE CODE COURSE TITLE NO. OF LECTURES / WEEK NO. OF CREDITS FYBSc / BA 1 SMAT 101/ AMAT 101 Calculus I SMAT 102 Algebra I SMAT1 PR1 /AMAT1 PR1 SMAT 201/ AMAT 201 Practical-I (Based on SMAT 101/AMAT 101, SMAT 102) 3 2 Calculus II SMAT 202 Algebra II SMAT2 PR2 /AMAT2 PR2 Practical-II (Based on SMAT 201/AMAT 201, SMAT 202) 3 2 2

3 F.Y.B.Sc./B.A. Introduction : Mathematics pervades all aspects of life, whether at home, in civic life or in the workplace. It has been central to nearly all major scientific and technological advances. Many of the developments and decisions made in our community rely to an extent on the use of mathematics. Besides foundation skills and knowledge in mathematics for all citizen in the society, it is important to widen mathematical experience for those who are mathematically inclined. Aims : (a) Giving students sufficient knowledge of fundamental principles, methods and a clear perception of boundless power of mathematical ideas and tools and know how to use them by analysing, modeling, solving and interpreting. (b) Reflecting on the broad nature of the subject and developing mathematical tools for continuing further study in various fields of science. (c) Enhancing student s overall development and to equip them with mathematical modeling abilities, problem solving skills, creative talent and power of communication necessary for various kinds of employment. (d) A student should get adequate exposure to global and local concerns by looking at many aspects of mathematical Sciences. Outcomes : (a) Student s Knowledge and skills will get enhanced and they will get confidence and interest in mathematics, so that they can master mathematics effectively and will be able to formulate and solve problems from mathematical perspective. (b) Student s thinking ability and attitude will change towards learning mathematics and practicals will improve their logical and analytical thinking. 3

4 SEMESTER I CALCULUS I Course Description: We begin with a brief introduction of limits, continuity and differentiability which will enable students to form and solve differentiable equations. Variety of applications of differential equations will be demonstrated for real world problems. Next we will introduce real numbers and properties which will help students to understand the origin of number system. Basic theorems of real analysis like Archimedean property, Hausdorff property with applications will be introduced. After this we start with sequence of real numbers and concept of convergent sequences that will help students understand and solve problems which are widely prevalent in all branches of science. Syllabus Unit 1: Differential Equations (1) Solutions of homogeneous and non-homogeneous differential equations of first order and first degree, Notion of partial derivative, solving exact differential equations. (2) Rules for finding integrating factor (I.F) (without proof ) for non-exact equations such as: (i) 1 M x+n y (ii) 1 M x N y is an I.F if M x + N y = 0 and M dx + N dy is homogeneous. is an I.F if M x N y = 0 and M dx + N dy is of the type f 1 (xy)ydx + f 2 (xy)xdy. (iii) e R (f(x)dx) is an I.F if N = 0 and 1 ( M N ) is a function of x alone say f (x). N y x (iv) e R (f(y)dy) 1 is an I.F if M = 0 and ( N M ) is a function of y alone say f (y). M x y (3) Finding solutions of first order differential equations of the type dy + P (x)y = Q(x)y n dx for n 0. Applications to orthogonal trajectories, population growth, and finding the current at a given time. 4

5 Unit 2: Real Numbers (1) Real number system R and order properties of R, Elementary consequences of these properties including AM-GM inequality. (2) Absolute value function (modulus) on R, Examples and basic properties. (3) Triangle inequality, Intervals and neighborhoods. (4) Bounded sets of real numbers, Supremum(l.u.b) and Infimum (g.l.b), l.u.b and g.l.b property and its applications (5) Archimedean property and its applications like Density theorem, nested interval theorem, existence of square root of 2 Unit 3: Sequences (1) Definition of a sequence and examples, Convergence and divergence of sequences, Convergent sequence is bounded, Uniqueness of limit if it exists. Examples on convergence of a sequence using -n 0 definition. (2) Sandwich theorem, Algebra of convergent sequences, Examples. (3) Bounded sequences, Monotone sequences and their convergence. (4) Standard examples such as a n, an, (1 + 1 ) n, n! n 1! 2! n! , a n (a > 0), n n. (5) Cauchy sequences and their convergence, subsequences and their convergence, Bolzano- Weierstrass theorem. Practicals for Calculus-I (1) Solving exact and non-exact differential equations. (2) Solving linear differential equations, Bernoulli s differential equations and its applications. (3) Problems based on absolute value and properties of R. (4) Problems on bounded sets and Archimedean property. (5) Problems on convergent sequences. (6) Problems based on subsequences and Cauchy sequences. (7) Definitions and statements of important theorems. 5

6 References [1] Dennis Zill, A first course in differential equations with modeling applications, Brooks/Cole ninth edition. [2] R.G. Bartle and D.R. Sherbert, Introduction to real analysis, John Wiley and Sons. [3] Ajit Kumar and S. Kumaresan, A basic course in real analysis, CRC press Additional References (1) G.F. Simmons, Differential equations with applications and historical notes, McGraw Hill. (2) R.R. Goldberg, Method of real analysis, Oxford and IBH, (3) T.M. Apostol, Calculus Volume I, Wiley and Sons (Asia). (4) K.G. Binmore, Mathematical Analysis, Cambridge university press,

7 ALGEBRA-I Course Description: The aim of this course is to introduce students to basic concepts like sets, relations, equivalence relations and functions, etc. It will make students learn different techniques of proving theorems, lemmas using induction, proof by contradiction etc. We also equip them with integers, division algorithm, congruences and its applications. Syllabus Unit-1: Sets and functions (1) Negation of a statement, use of quantifiers, sets, union and intersection of sets, complement of a set, De Morgan s law, Cartesian product of sets. (2) Definition of a function; domain, co-domain and range of a function, composite functions, examples, Graph of a function, Injective, surjective, bijective functions; composite of injective, surjective, bijective functions when defined. (3) Invertible functions, bijective functions are invertible and conversely. Examples of functions including constant, identity, projection, inclusion. (4) Image and inverse image of a set under f interrelated with union, intersection and complement. Finite and infinite sets. Countable set and its examples such as Z, Q. Uncountable set and its examples. Unit-2: Integers and divisibility (1) Well-ordering property, First and second principle of mathematical induction as a consequence of well-ordering property (2) Divisibility in integers, division algorithm, existence & uniqueness of greatest common divisor(g.c.d.) and least common multiple (l.c.m.) and their basic properties. Bezout s identity and its applications. (3) Euclidean algorithm, Primes, Euclid s lemma, Fundamental theorem of arithmetic, The set of primes is infinite. (4) The necessary and sufficient condition to have a solution for the linear Diophantine equation ax + by = c. Solving of linear Diophantine equation with examples. 7

8 Unit-3: Theory of congruences (1) Equivalence relation, equivalence classes and properties, Definition of a partition, every partition gives an equivalence relation and vice versa. Congruences, definition and elementary properties, Congruence is an equivalence relation on Z, residue classes and partition of Z, addition modulo n, multiplication modulo n, examples (2) Linear congruences. Chinese remainder theorem and its applications (3) Euler s φ function, Euler s theorem, Fermat s little theorem, Wilson s theorem and their applications. Practicals for Algebra-I (1) Functions(image and inverse image), injective, surjective, bijective functions, finding inverses of bijective functions. (2) Problems on countability. (3) Problems on mathematical induction, Euclidean algorithm in Z. (4) Problems on fundamental theorem of arithmetic and solving linear Diophantine equations. (5) Problems on congruences, equivalence relation and Chinese remainder theorem. (6) Problems on Euler s φ function, Fermat s little theorem, Wilson s theorem. References [1] S. Kumaresan, Ajit Kumar and Bhaba Kumar Sarma, A foundation course in Mathematics, 2018 edition, Narosa publication house. [2] David M. Burton, Elementary number theory, seventh edition, Tata McGraw-Hill edition. Additional references (1) Ivan Niven, Herbert S. Zuckerman, Introduction to the theory of numbers, fifth edition, Wiley eastern limited. (2) R.G. Bartle and D.R. Sherbert, Introduction to real analysis, third edition, John Wiley and Sons. (3) Jones and Jones, Elementary number theory, second edition, Springer (4) I.S. Luthar, Sets, functions and numbers, 2005 edition, Narosa publishing house. (5) Thomas Koshy, Elementary number theory with applications, Academic press 8

9 SEMESTER II CALCULUS II Course Description: The aim of this course is to expose students to the beauty of limits, continuity and the concept of differentiation. The first unit is based on limits and continuity in which students learn the definition of continuity and sequential continuity and the equivalence between them. Problems based on these concepts are solved rigourously. The next unit is based on differentiability. Here students understand the notion of differentiation of a real valued function and mean value theorems. In the last section the emphasis is on applications of differentiability. Syllabus Unit 1: Limits and continuity (1) -δ definition of limit of a (real valued) function, Right hand and Left hand limits, Uniqueness of limit when it exists, Algebra of limit of a function, Sandwich theorem. (2) -δ definition of continuity of a (real valued) function, examples, Sequential continuity. (3) Alebra of continuous functions, Continuity of f when f is continuous. Continuity of composition of two continuous functions. (4) Examples of discontinuous functions and continuity of constant function, identity function, trigonometric functions, polynomial functions etc. (5) Intermediate value theorem and its applications, A continuous function on a closed and bounded interval is bounded and attains its bounds and its consequences. 9

10 Unit 2: Differentiability (1) Differentiation of a real-valued function, examples of differentiable and non-differentiable functions, differentiability implies continuity, Algebra of differentiable functions,derivative of a inverse function. (2) Chain-Rule, Higher order derivatives, Leibnitz rule, L Hospital s rule, examples of indeterminate form. (3) Rolle s theorem, Lagrange s and Cauchy s mean value theorem, their applications and examples. Unit 3: Applications of Differentiation (1) Taylor s theorem and its applications. (2) Definition of local maximum and local minimum, necessary condition, stationary points, first and second derivative test, examples, Graphing of functions using first and second derivatives. (3) Application to economics and commerce, Concave, convex functions, points of inflections. Practicals for Calculus-II (1) Problems on limits and continuity using definiton (2) Problems on sandwich theorem and intermediate value theorem. (3) Problems on differentiability and Leibnitz rule, L Hospital s rule (4) Problems based on mean value theorems. (5) Problems on local maxima and minima. (6) Concavity of curves with graphs. References [1] R.G. Bartle and D.R. Sherbert, Introduction to real analysis, John Wiley and Sons. [2] Ajit Kumar and S. Kumaresan, A basic course in real analysis, CRC press [3] James Stewart, Calculus: early transcendentals, Cengage, 7th edition

11 Additional References (1) Strauss, Bradley and Smith, Calculus, Pearson 3rd edition, (2) R.R. Goldberg, Method of real analysis, Oxford and IBH, (3) T.M. Apostol, Calculus Volume I, Wiley and Sons (Asia). (4) K.G. Binmore, Mathematical Analysis, Cambridge university press,

12 ALGEBRA-II Course Description: The aim of this course is to introduce System of linear equations and matrices and to understand polynomials over R and permutations of a set. The first unit is devoted to system of linear equations. We then proceed to introduce permutations, symmetries, cycles and give its applicaions. In the last unit we introduce polynomials over set of reals and complex numbers. We also introduce gcd of two polynomials over R, Euclidean algorithm and solve problems based on that. Different techniques using rational root theorem will enable students to find roots of a polynomial. Unit-1: System of linear equations and matrices (1) System of homogeneous and non-homogeneous linear equations, the solution of system of m homogeneous linear equations in n unknowns by elimination and its geometrical interpretation. (2) Definition of an n-tuple of real numbers, sum of two n-tuples and scalar multiplication of an n-tuple. (3) Matrices with real entries; addition, scalar multiplication and multiplication of matrices; transpose of a matrix, types of matrices: zero matrix, identity matrix, diagonal matrices, triangular matrices, symmetric matrices, skew-symmetric matrices, invertible matrices, identities such as (AB) t = B t A t, (AB) 1 = B 1 A 1 (4) System of linear equations in matrix form, elementary row operations, row echelon form, Gauss elimination method, the system of m homogeneous linear equations in n unknowns has a non-trivial solution if m < n. Unit-2: Permutations (1) Definition of a permutation of a set, Set of all permutations of the set {1, 2,..., n} i.e. S n and its cardinality, Symmetries of an equilateral triangle, square, rectangle. (2) Cycles, Composition of permutations, properties of permutations such as every permutation of a finite set can be written as a cycle or a product of disjoint cycles, disjoint cycles commute. (3) Transpositions, Any permutation can be expressed as a product of transpositions, order of a permutation, sign of a permutation. 12

13 Unit-3: Polynomials (1) Definition of a polynomial, polynomials over the field F where F = Q, R or C, Algebra of polynomials, degree of a polynomial, basic properties. (2) Division algorithm for polynomials over R, gcd of two polynomials and its basic properties, Euclidean algorithm and its applications, roots of a polynomial, relation between roots and coefficients, multiplicity of a root, remainder theorem, factor theorem. (3) A polynomial of degree n has atmost n roots, Complex roots of a polynomial in R[X ] occur in conjugate pairs, statement of Fundamental theorem of algebra, a polynomial of degree n. (4) Rational root theorem and its consequences such as p is an irrational number when p is a prime number, roots of unity, sum of all the roos of unity Practicals for Algebra-II (1) Problems of solving homogeneous system of m equations and n unknowns by elimination, problems on row echelon form (2) Solving a system Ax = b by Gauss elimination, finding inverse of a matrix if it exists, solutions of system of linear equations. (3) permutations of a finite set, Symmetries, cycles, compositions of permutations. (4) Permutation as a product of 2-cycles, order and sign of a permutation. (5) Problems on division algorithm and gcd of two polynomials. (6) Problems based on factor theorem, remainder theorem and rational root theorem. References [1] S. Kumaresan, Linear algebra, a geometric approach first edition, Prentice hall of India, [2] Joseph A. Gallian, Contemporary abstract algebra, fourth edition, Narosa publications. [3] John Fraleigh, A first course in abstract algebra, seventh edition, Pearson, Additional References (1) Norman L. Biggs, Discrete mathematics, second edition, Oxford university press. (2) I.N. Herstein, Topics in algebra, second edition, Wiley India edition. (3) Serge Lang, Introduction to linear algebra, second edition, Springer. 13

14 Exam pattern (1) Semester End Exam (60 marks). (2) CAI : 20 marks (Test). (3) CAII : 20 marks Assignment containing 5 problems. (4) Practical exam 50( ) marks. 14

15 Paper Pattern (1) CA I : Problem solving test of 20 marks. (2) CA II(Assignment): This is given to a group of students. The Pattern is: (i) 5 Question based on Theorems OR (ii) 5 question based on Problems. Semester End Exam Pattern Based on Unit I (Q.1/A) Attempt any 1 out of 2. 8 Marks each. (Q.1/B) Attempt any 2 out of 3. 4 Marks each. Based on Unit II (Q.2/A) Attempt any 1 out of 2. 8 Marks each. (Q.2/B) Attempt any 2 out of 3. 4 Marks each Based on Unit III (Q.3/A) Attempt any 1 out of 2. 8 Marks each. (Q.3/B) Attempt any 2 out of 3. 4 Marks each Based on Full Syllabus (Q.4) Attempt any 2 out of 3. 6 Marks each. Practical Exam Pattern (Q.1) 10 Marks will be for question based on definitions, statements, multiple choice, True False, etc. (Q.2) 6 Marks For Journal and 6 Marks for Viva. (Q.3) 18 Marks Descriptive type where students have to solve 3 questions of 6 marks each out of 4 given choices. 15

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