### Academics

#### Course Outline

**B.Tech Mathematics Outline Syllabi **

**Core Course**

**MATHEMATICS I (Credit: 4) 1 ^{st} Semester Sub.Code:- **14B11MA111

Calculus and Analytical Geometry: Partial differentiation, Taylors series for functions of two variables, maxima and minima of fuctions of two variables. Jacobians. Double integrals, Line and surface integrals. Gradient, divergence and curl. Normal and tangent to a surface. Gauss and Stokes’ theorems.

Differential Equations I: Differential Equations with constant coefficients. Cauchy’s -Euler equation. Equations of the form y” = f(y).Simple applications. Laplace Transform. Inverse Laplace transforms. Dirac delta and unit step function. Solution of initial value problems.

Matrices: Linear dependence & independence of rows, row echelon form, Rank. Solution of system of equations by Guass elimination. Eigenvalues and vectors of a square matrix, symmetric matrices, Reduction to diagonal form, Quadratic forms.

**BASIC MATHEMATICS I (Credit: 4) **

Sets, Relations and Functions: Sets and their representation. Union, intersection and compliment. Mapping or function. One-one, onto mappings. Inverse and composite mappings.

Complex Numbers: Definition and geometrical representation. Algebra. Complex conjugate. Modulus and amplitude. Polar form. DeMoivre’s theorem. Roots of complex numbers. Simple functions.

Differential Calculus: Basic concept of limit and continuity. Derivative. Rules of differentiation. Tangent to a curve. Taylor’s series. Maxima and minima.

Integral Calculus: Antiderivative. Fundamental theorem of calculus (statement only). Integrals of elementary functions. Substitution and partial fractions. Definite integral as a limit of sum. Properties of definite integrals. Application to areas and lengths.

Matrices and Determinants: Algebra of matrices. Determinant of a square matrix. Properties of determinants. Some simple type of matrices. Inverse of a matrix. Solution of equations.

Two dimensional coordinate Geometry: Cartesian coordinate system. Distance between two points. Equation of line in different forms. Equations of circle, ellipse and parabola. Equation of a tangent to a curve. Area of a triangle.

**MATHEMATICS II (Credit: 4) 2 ^{nd} Semester Sub.Code:- **14B11MA201

Sequence and Series: Comparison test, Ratio test, Integral test, Raabe’s test, Cauchy nth root test, Logarithmic test. Alternating Series, Conditional & Absolute Convergence, Uniform Convergence.

Differential Equations II: Second order linear differential equations, Change of dependent and independent variables, variation of parameters. Solution in series- Bessel and Legendre functions, Orthogonality, Generating functions and recurrence relations (without proofs). Classification of Second order partial differential equations.

Method of separation of variable. One dimensional wave equation, heat conduction equation and Laplace equation.

Functions of a complex variable: Analytic functions Cauchy-Riemann equations. Poles and singularities. Complex Integration. Cauchy’s Integral theorem. Couchy’s Integral Formula. Taylor’s and Laurent’s series. Cauchy residue theorem and applications, Bilinear transformations.

**BASIC MATHEMATICS II (Credit: 4)**

Sequence and Series: Convergence and divergence. Simple tests for convergence. Absolute convergence. Fourier series. Vectors and Coordinate Geometry (3D): Vectors and their algebra. Simple applications to geometry and mechanics. Unit vectors, vectors *i, j* and *k*. Components of a vector. Position vector. Direction cosines and direction ratios. Dot and cross products. Projection of a vector on another. Distance between two points. Equations of a line, plane and sphere. Intersections. Distance between two points. Shortest distance between lines.

Calculus of two or more variables: Partial differentiation. Taylor’s series. Differentiation of a vector. Tangent to a curve. Gradient of a scalar. Tangent to a surface. Integration of a vector. Line integral. Double integral. Change to polar coordinates. Applications.

Elementary Differential Equations: Definitions of order, degree, linear, nonlinear, homogeneous and nonhomogeneous. Solution of first order equations. Complementary function and particular integral. Initial and boundary value problems. Linear differential equations with constant coefficients. Cauchy-Euler equation. Solution in series.

Basic Statistics and Numerical Methods: Classification of data. Mean, mode, median and standard deviation. Method of least squares. Newton-Raphson method. Linear and quadratic interpolation. Simpson’s rule. Runge-Kutta method.

**DISCRETE MATHEMATICS (Credit: 4) 2 ^{nd} Semester Sub.Code:- **14B11MA211

Relations and Logic: Review of relations, Partial Ordered relations, Hasse diagram, Lattices, Recursive functions, Recurrence relations, Solutions of recurrence relations by generating function and Z transform. Propositions- simple and compound. Basic logical operators. Implication. Truth tables. Tautologies and contradictions. Valid arguments and fallacy. Propositional functions and quantifiers.

Graph Theory: Graphs and related definitions, Subgraphs, isomorphism, paths and connectivity. Eulerian graph and Konigsberg problem. Hamiltonian graph. Labelled and weighted graphs. Trees. Graph colorings. Four color problem. Digraphs and related definitions. Rooted trees. Algebraic expressions and Polish notation. Sequential representation. Sequential representation. Adjacency matrix. Path matrix. Shortest path. Linked representation of directed graphs. Binary trees.

Algebraic Structures: Groups, order of group and its elements, Subgroups, Lagrange’s Theorem, Quotient groups, Rings, Integral domains, Fields.

Languages and Grammars: Languages, Regular Expressions, Grammars, Finite state Machine, Finite State Automata.

**PROBABILITY AND STATISTICS (Credit: 4) **

Classification of data, graphic and diagrammatic representation of data, measures of central tendency and dispersion i.e. mean and standard deviation, measures of skewness and kurtosis.

Sample space and events. Permutations and combinations. Probability of an event. Axioms of probability. Equiprobable spaces. Conditional probability. Multiplication and addition theorems. Baye’s theorem. Independent events.

Random Variable. Discrete and continuous distributions. Mean and variance of a random variable. Binomial, Uniform, Normal and Poisson distributions.

Elementary sampling theory, Distribution of Means and Proportions. Statistical decision theory based on large sample theory. Test of hypothesis and significance. Test based on Exact (Small) Sampling- Chi-square test, t test and F test. Curve fitting by the method of least squares. Correlation and regression.

Time Series Analysis and Moving Average.

**PROBABILITY THEORY AND RANDOM PROCESSES (Credit: 4) 4rth Semester Sub.Code:- **14B11MA411

Probability: Three basic approaches to probability, Conditional probability, total probability theorem, Bayes theorem.

Random variables: One dimensional random variables (discrete and continuous), distribution of a random variable (density function and cdf). MGF and Characteristic function of a random variable and its utility. Bivariate random variable, joint, marginal and conditional distributions, covariance and correlation.

Probability Distributions: Bernoulli, binomial, Poisson, negative binomial, geometric distributions. Uniform, exponential, normal, gamma, Earlang, and Weibull distributions.

Reliability: Concept of reliability, reliability function, hazard rate function, mean time to failure (MTTF). Reliability of series, parallel, series-parallel, parallel-series systems.

Random processes: Introduction, Statistical description of random processes, Markov processes, processes with independent increments. Average values of random processes. Stationary (strict sense and wide sense) processes, and computation of their averages. Random walk, wiener process. Semi-random telegraph signal and random telegraph signal process. Properties of autocorrelation function, ergodic processes. Power spectral density function and its properties. Poisson processes. Markov chains and their TPM (transition probability matrix).

__Elective Courses__

**APPLIED LINEAR ALGEBRA (Credit: 3) **

Vector Space: Subspaces, linear dependence and independence, span of a set, bases and dimension. Direct sum and complement.

Linear Transformation: Definition. Algebra of linear transformations, isomorphism, representation by matrices. Rank and nullity theorem and its consequences. Change of basis. Inverse of a linear transformation, linear functional, Transpose. Application to system of linear equations.

Inner Product and Orthogonality: Metric and normed spaces. Inner product and inner product spaces. Orthogonality. Orthonormal basis. Gram-Schmidt orthogonalization. Expansion. Orthogonal and unitary matrices.

Eigenvalues and Vectors: Modal matrix and diagonalization. Similarity Transformation. Powers and functions of matrices. Eigensystems of real symmetric, orthogonal, Hermitian and unitary matrices. Quadratic forms. Positive definite matrices. Computation of eigenvalues. Norm of a matrix. Condition number. Application to solving ordinary differential equations.

**APPLIED NUMERICAL METHODS (Credit: 3) **

Numerical Linear Algebra: Gauss-elimination and LU-Decomposition. Iterative methods: Gauss Seidel and successive-overrelaxation. Power method for largest eigenvalue. Jacobi method for real symmetric matrices.

Interpolation and Approximation: Interpolating polynomial. Lagrange formula with error. Formulae for equispaced points. Divided differences. Hermite interpolation. Least square approximation. Bezier methods.

Numerical Differentiation and quadrature: Approximation of derivatives. Newton-Cote formulae. Gauss-Legendre quadrature formulae. Double integration.

Non-linear Algebraic Equations: Iterative methods for one or more nonlinear equations with convergence.

Numerical Solutions of ODE and PDE : Runge-Kutta and predictor corrector methods for IVPs. Finite difference methods for BVPs. Shooting methods, Numerical solutions of parabolic and elliptic partial differential equations.

**OPERATIONS RESEARCH (Credit: 3) 4rth Semester Sub.Code:-** 14B1AME414

Preliminaries: Introduction, Operations Research Models, Phases and Scope of O.R. Studies.

Linear Programming Problems (LPP): Convex Sets, Formulation of LPP, Graphical Solutions, Simplex Method, Big-M Method, Two Phase Method, Special Cases in Simplex Method.

Duality and Sensitivity Analysis: Primal-Dual Relationship, Duality, Dual Simplex Method, Sensitivity Analysis..

Transportation Problems: Introduction, Matrix Form, Applications, Basic Feasible Solution- North West Corner Rule, Least Cost Method, Vogel’s Approximation Method. Degeneracy, Resolution on Degeneracy, Optimal Solution, Maximization TP Model

Assignment Problems: Definition, Hungarian Method, Traveling Salesmen Problems.

Integer Linear Programming Problems: Pure and Mixed Integer Linear Programming Problems, Cutting Plane Method, Branch and Bound Method.

Introduction to NLP: Convex Functions, Graphical Solution. Unconstraint Problem-Necessary and Sufficient Condition for extreme points, Quadratic Programming -Wolfe’s Method. Constrained Problems – Lagrange Method for Equality Constraints, Kuhn-Tucker Conditions for Inequality Constraints.