Course Code: MCA113 |
Course Title: Foundation of mathematics (4 Credits) |
Course Contents
Unit-1 Set Theory
Sets and Their Representations, The Empty Set, Finite and Infinite Sets, Equal and Equivalent Sets, Subsets, Power Set, Universal Set, Venn Diagrams, Complement of a Set, Operations on Sets, Applications of Sets, Cartesian Product of Sets.
Unit-2 Limits and Continuity
The Real Number System, the Concept of Limit, Concept of Continuity
Unit-3 Differential calculus-I
Differentiation of Powers of x, Differentiation of ex and log x, Differentiation of Trigonometric Functions, Rules for Finding Derivatives, Different types of Differentiation, Logarithmic Differentiation, Differentiation by Substitution, Differentiation of Implicit Functions, Differentiation from Parametric Equation, Differentiation from First Principles.
Unit-4 Differential calculus-II
Successive Differentiation, Leibnitz’s theorem (without proof), Lagrange’s Theorem, Cauchy Mean value Theorem, Taylor’s theorem (without proof), Asymptotes.
Unit-5 Integral calculus-I
Integration of Standard Functions, Rules of Integration, More Formulas in Integration, Definite Integrals
Unit-6 Integral calculus-II
Reduction Formulae of trigonometric functions, Properties of definite Integral, Applications to length, area, volume, surface of revolution, definition of improper integrals, Beta- Gamma functions.
Unit-7 Calculus of Functions of several variables
Partial derivatives, Chain Rule, Differentiation of implicit functions, Exact differentials, Maxima, Minima and saddle points, methods of Lagrange multipliers. Differential under Integral sign, Jacobians and transformations of coordinates, Double and triple integrals Simple applications to areas, volumes etc
Unit-8 Vector calculus-I
Vector Algebra: Definition of a Vector, Addition and Subtraction, Components, Physical examples
Unit-9 Vector calculus-II
Vector Products: Scalar and Vector products including a brief introduction to determinants, triple products, Geometrical applications. Differentiation and Integration of a Vector functions Serret - Frenet formulas.
Unit-10 Vector calculus-III
Vector Analysis: Scalar and vector fields, Curves, arc length, tangent, Normal, directional derivatives, Gradiant of scalar field, divergence and curl of vector field, line integral (independent path)
Unit-11 Vector calculus-IV
Green’s theorem, Divergence theorem, and stroke’s theorem, (without proof) with physical examples, surface Integrals.
Unit-12 Matrices
Elementary row and column transformation, linear dependence, rank of a matrix, consistency of system of linear equations, solution of linear system of equations, characteristic equations, Cayley Hamilton theorem, eigen values and eigen vectors, diagonalization, complex matrices
Unit-13 Complex variables
Curves and Regions in the complex Plane, Complex functions, Limits, Derivatives, Analytic function, Cauchy-Riemann equations, Laplace’s equation, Complex line Integral, cauchy’s Integral Theorem, Cauchy’s Integral Formula, Power series, Taylors series, Laurent series. Methods of obtaining zeros, Singularities, residues, Residue Theorem.
Unit-14 Mathematical Logics
Statements, Basic Logical Connectives, Conjunction, Disjunction, Negation, Negation of Compound Statements, Truth Tables, Tautologies, Logical Equivalence, Applications.