10B11MA111 MATHEMATICS I (Credit: 4) Calculus and Analytical Geometry: Partial differentiation, Taylors series for fu nctions of two variables, maxima and minima of fuctions of two variables. Jacobi ans. 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. Cau chy chys -Euler equation. Equations of the form y = f(y).Simple applications. Laplace Transform. Inverse Laplace transforms. Dirac delta and unit step function. Solut ion of initial value problems. Matrices: Linear dependence & independence of rows, row echelon form, Rank. Solu tion of system of equations by Guass elimination. Eigenvalues and vectors of a s quare matrix, symmetric matrices, Reduction to diagonal form, Quadratic forms.
10B11MA112 BASIC MATHEMATICS I (Credit: 4) Sets, Relations and Functions: Sets and their representation. Union, intersectio n and compliment. Mapping or function. One-one, onto mappings. Inverse and compo site mappings. Complex Numbers: Definition and geometrical representation. Algebra. Complex con jugate. Modulus and amplitude. Polar form. DeMoivre s theorem. Roots of complex nu mbers. 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 on ly). Integrals of elementary functions. Substitution and partial fractions. Defi nite integral as a limit of sum. Properties of definite integrals. Application t o 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. S olution of equations. Two dimensional coordinate Geometry: Cartesian coordinate system. Distance betwe en 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.
10B11MA201 MATHEMATICS II (Credit: 4) Sequence and Series: Comparison test, Ratio test, Integral test, Raabe s test, Cau chy nth root test, Logarithmic test. Alternating Series, Conditional & Absolute Convergence, Uniform Convergence. Differential Equations II: Second order linear differential equations, equations, Change o f dependent and independent variables, variation of parameters. Solution in seri es- Bessel and Legendre functions, Orthogonality, Generating functions and recur rence relations (without proofs). Classification of Second order partial differe ntial equations. Method of separation of variable. One dimensional wave equation, heat conduction equation and Laplace equation. Functions of a complex variable: Analytic Analytic functions Cauchy-Riemann equations. P oles and singularities. Complex Integration. Cauchy s Integral theorem. Couchy Couchy s Int
egral Formula. Taylors and Laurents series. ns, Bilinear transformations.
Cauchy residue theorem and applicatio
10B11MA212 BASIC MATHEMATICS II (Credit: 4) Sequence and Series: Convergence and divergence. Simple tests for convergence. Absolute convergence. Fourier series. Vectors and Coordinate Geometry (3D): Vect ors and their algebra. Simple applications to geometry and mechanics. Unit vecto rs, vectors i, j and k. Components of a vector. Position vector. Direction cosin es and direction ratios. Dot and cross products. Projection of a vector on anoth er. Distance between two points. Equations of a line, plane and sphere. Intersec tions. Distance between two points. Shortest distance between lines. Calculus of two or more variables: Partial differentiation. Taylor s series. Diffe rentiation of a vector. Tangent to a curve. Gradient of a scalar. Tangent to a s urface. Integration of a vector. Line integral. Double integral. Change to polar coordinates. Applications. Elementary Differential Equations: Definitions of order, degree, linear, nonline ar, homogeneous and nonhomogeneous. Solution of first order equations. Complemen tary function and particular integral. Initial and boundary value problems. Line ar differential equations with constant coefficients. Cauchy-Euler equation. Sol ution in series. Basic Statistics and Numerical Methods: Classification of data. Mean, mode, medi an and standard deviation. Method of least squares. Newton-Raphson method. Linea r and quadratic interpolation. Simpson s rule. Runge-Kutta method.
10B11MA211 DISCRETE MATHEMATICS (Credit: 4) Relations and Logic: Review of relations, Partial Ordered relations, Hasse diagr am, Lattices, Recursive functions, Recurrence relations, Solutions of recurrence relations by generating function and Z transform. Propositions- simple and comp ound. Basic logical operators. Implication. Truth tables. Tautologies and contra dictions. 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 r elated definitions. Rooted trees. Algebraic expressions and Polish notation. Seq uential 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, Lagran ges Theorem, Quotient groups, Rings, Integral domains, Fields. Languages and Grammars: Languages, Regular Expressions, Grammars, Finite state M achine, Finite State Automata.
10B11MA311 PROBABILITY AND STATISTICS (Credit: 4) Classification of data, graphic and diagrammatic representation of data, measure s 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. Multiplic
ation and addition theorems. Bayes theorem. Independent events. Random Variable. Discrete and continuous distributions. Mean and variance of a r andom variable. Binomial, Uniform, Normal and Poisson distributions. Elementary sampling theory, Distribution of Means and Proportions. Statistical d ecision 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. Curv e fitting by the method of least squares. Correlation and regression. Time Series Analysis and Moving Average.
10B11MA411 PROBABILITY THEORY AND RANDOM PROCESSES (Credit: 4) Probability: Three basic approaches to probability, Conditional probability, tot al probability theorem, Bayes theorem. Random variables: One dimensional random variables (discrete and continuous), di stribution of a random variable (density function and cdf). MGF and Characterist ic function of a random variable and its utility. Bivariate random variable, joi nt, marginal and conditional distributions, covariance and correlation. Probability Distributions: Bernoulli, binomial, Poisson, negative binomial, geom etric distributions. Uniform, exponential, normal, gamma, Earlang, and Weibull d istributions. 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, Mar kov processes, processes with independent increments. Average values of random p rocesses. Stationary (strict sense and wide sense) processes, and computation of their averages. Random walk, wiener process. Semi-random telegraph signal and r andom 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).