(For I B.Tech Common to all Branches)                         4-0-6

UNIT NO.

 Content

Rolle’s theorem – Lagrange’s Mean Value theorem - Cauchy’s Mean value theorem – Generalized Mean Value theorem (Taylor’s Theorem). , Radius, Center and Circle of Curvature – Evolutes and Envelopes.

Functions of several variables – limit and continuity - partial differentiation – Chain rule – Total derivative - Euler’s theorem, Jacobian – Functional dependence.  Maxima and Minima of functions of two variables with and without constraints,

Curve tracing – Cartesian, polar and Parametric curves.  Applications of integration to lengths, volumes and surface areas in Cartesian and Polar coordinates.

Differential Equations of first order and first degree – formation, Exact, Linear and Bernoulli. Applications to Newton’s Law of cooling, Law of natural growth and decay, Orthogonal trajectories. Homogeneous and Non-Homogeneous LDE with constant coefficients of second and higher orders.  Particular integral when RHS term of the type e^ax , sin ax, cos ax, polynomial in x, e^ax  V(x), xV(x).  Method of variation of parameters.  Applications of II order Linear ODEs Mass Spring System, damped, critical and undamped motion.

Laplace transform of standard functions – Inverse transform – Linearity – first shifting.  Transforms of derivatives and integrals – Unit step function – second shifting theorem – Dirac’s delta function – Response of a damped vibrating system to single square wave and to a unit impulse function - Differentiation and integration of transforms – convolution theorem – Transform of periodic function, Application to ordinary differential equations and simultaneous equations.

Multiple integrals.  Double and triple integrals change of variables change of order of integration. Pappus’ Theorems. Moment of Inertia, Center of Gravity

Gradients, divergence, Curl and their related properties of sums.  Products, Laplacian and second order operators. Vector integral Calculus:

 Line integral – work done  - Potential function, area - Surface and volume integrals.  Green’s theorem, Stoke’s Theorem and Gauss’s Gauss divergence Theorem.  Verification of Green’s stroke’s and Gauss’s theorems.  

4-0-4

UNIT NO.

 Content

Matrices: Elementary row transformations – Rank – Normal form – Echelon form – Consistency – Solution of system of simultaneous linear homogeneous and Non-homogeneous equations. LU Decomposiion method.

Eigen values, Eigen vectors – properties – Cayley- Hamilton Theorem – Inverse and powers of a matrix by Cayley-Hamilton theorem – Diagonolization of matrix. Calculation of powers of a matrix – Modal and spectral matrices. Real matrices – Symmetric, skew-symmetric, orthogonal, Linear Transformation – Orthogonal Transformation. Complex matrices: Hermitian, Skew-Hermitian and Unitary – Eigen values and Eigen vectors of complex matrices and their properties.

Quadratic forms – Reduction of quadratic form to canonical form – Rank – Positive, negative definite – semi definite – index – signature – Sylvester law.

Fourier Series: Determination of Fourier coefficients – Fourier series – even and odd functions – Fourier series in an arbitrary interval – even and odd periodic continuation – Half-range Fourier sine and cosine expansions.

Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions – solutions of first order linear (Lagrange) equation and nonlinear (standard type) equations. Method of separation of variables – Classification of second order linear partial Differential equations, solutions of one-dimensional heat equation, wave equation and two-dimensional Laplace’s equation under initial and boundary conditions.

Fourier integral theorem – Fourier sine and cosine integrals. Fourier transform – Fourier sine and cosine transforms – properties – inverse transforms – Finite Fourier transforms.

Z-transform – inverse z-transform – properties – Damping rule – Shifting rule – Initial and final value theorems. Convolution theorem – Solution of difference equation by z-transforms.

Wavelets – The Haar wav lets – A wavelet expansion – Multiresolution analysis with Haar wavelets – General construction of wavelets and multi-resolution analysis – Shannon wavelets.

4-0-4

UNIT NO.

 Content

Special functions: Gamma and Beta Functions – Their properties – evaluation of improper integrals. Bessel functions – properties – Recurrence relations – Orthogonality. Legendre polynomials – properties – Rodrigue’s formula – Recurrence relations – Orthogonality.

Functions of a complex variable – Continuity – Differentiability – Analyticity – Properties – Cauchy-Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions – Milne – Thompson method.

Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties – General power Z^c (c is complex), principal value.

Complex integration: Line integral – evaluation along a path and by indefinite integration – Cauchy’s integral theorem – Cauchy’s integral formula – Generalized integral formula.

Complex power series: Radius of convergence – Expansion in Taylor’s series, Maclaurin’s series and Laurent series. Singular point – Isolated singular point – pole of order m – essential singularity.

Residue – Evaluation of residue by formula and by Laurent series – Residue theorem. Evaluation of integrals of the type (a) Improper real integrals ò f (x) dx (b) ò f (cosq, sinq) dq (c) ò  e^imx f (x) dx ,    (d) Integrals by indentation.

Argument principle – Rouche’s theorem – determination of number of zeros of complex polynomials – Maximum Modulus principle – Fundamental theorem of Algebra, Liouville’s Theorem.

Conformal mapping: Transformation by e^z, lnz, z², zⁿ (n positive integer),  Sin z, Cos z, z+a/z. Translation, rotation, inversion and bilinear transformation – fixed point – cross ratio – properties – invariance of circles and cross ratio – determination of bilinear transformation mapping 3 given points.

4-0-4

UNIT NO.

CONTENT

Sample space - events – probability – The axioms of probability – Some elementary theorems – conditional probability  - Baye’s theorem.

Random variables - Discrete and continuous distribution – Distribution function  –  Distributions – Binomial,  Poisson, and Normal distribution  - related properties.

Two-dimensional random variables – marginal , conditional distributions – discrete and continuous – Moment generating function of standard distributions – Characteristic functions

Point estimation – Interval estimation – Bayesian estimation – Tests of Hypothesis – Means and proportions – Hypothesis Concerning one and two means -- Type I and Type II  errors – One tail and  two-tail tests –OC curves - tests of  significance –  Student t-test , F- test, c² test, Estimation of proportions

Point estimation – Interval estimation – Bayesian estimation – Statistical Quality Control –UCL-LCL –p-charts –XBar-charts

Tests of Hypothesis – Means and proportions – Hypothesis Concerning one and two means -- Type I and Type II  errors – One tail and  two-tail tests – tests of  significance –  Student t-test , F- test, c² test, Estimation of proportions

Method of least squares -- Inferences based on the least squares estimation - curvilinear regression –– correlation for univariate and bivariate distributions:

Design of statistical experiments-Random block design –Latin squares –Orthogonal arrays.

( First year B.Tech Common to ECE, CSE, EEE)                       4-0-6

UNIT NO.

 Content

Matrices and Linear systems of equations: Elementary row transformations – Rank – Echelon form, Normal form – Solution of Linear systems – Direct Methods – LU decomposition – LU Decomposition from Gauss Elimination – Solution of tridiagonal Systems – Solution of Linear Systems.

Eigen values, Eigen vectors – properties – Cayley-Hamilton theorem – Inverse and powers of a matrix by Cayley-Hamilton theorem – Diagonolizaton of matrix. Calculation of powers of matrix – Modal and spectral matrices.

Real Matrices – Symmetric, skew-symmetric, orthogonal, Linear Transformation – Orthogonal Transformation. Complex matrices: Hermitian, Skew-Hermitian and Unitary – Eigen values and Eigen vectors of complex matrices and their properties.

Solution of Algebraic and Transcendental Equations: Introduction – The Bisection Method – The Method of False Position – The Iteration Method– Newton-Raphson Method.  Interpolation: Introduction – Errors in Polynomial Interpolation – Finite differences – Forward Differences – Backward differences – Central differences – Symbolic relations and separation of symbols – Differences of a polynomial – Newton’s formulae for interpolation – Central difference interpolation Formulae – Gauss’s Central Difference Formulae – Interpolation with unevenly spaced points – Lagrange’s Interpolation formula.

Numerical Differentiation and Integration: The Cubic Spline Method – Trapezoidal rule – Simpson’s 1/3 Rule – Simpson’s 3/8 Rule – Boole’s and Weddle’s Rules.

Numerical solution of Ordinary Differential equations: Solution by Taylor’s series – Picard’s method of successive Approximations- Euler’s Method – Runge-kutta Methods – Predictor-Corrector Methods – Adams-Moulton Method – Milne’s Method.

Fourier Series: Determination of Fourier coefficients – Fourier series – even and odd functions – Fourier series in an arbitrary interval – even and odd periodic continuation – Half-range Fourier sine and cosine expansions.  Fourier integral theorem (only statement) – Fourier sine and cosine integrals. Fourier transform – Fourier sine and cosine transforms – properties – inverse transforms – Finite Fourier transforms.

Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions – solutions of first order linear (Lagrange) equation and nonlinear (standard type) equations. Method of separation of variables. Z-transforms – inverse z-transform – properties – Damping rule – Shifting rule – Initial and final value theorems. Convolution theorem – Solution of difference equation by z-transforms.

4-0-4

UNIT

PRESENT

UNIT-1

Applied partial Differential Equations:  One-dimensional Heat equation Cartesian, cylindrical and spherical coordinates (problems having axi-symmetry)

UNIT-2

two-dimensional Laplace Equation in Cartesian, cylindrical and spherical coordinates (problems having axi-symmetry) – Analytical solution by separation of variables technique.

UNIT-3

Numerical solutions to Heat and Laplace Equations in Cartesian coordinates using finite – differences.

UNIT-4

Applied Statistics: Regression and correlation analysis – Method of Least squares – Curve fitting – Curvilinear Regression – Non-linear curves – correlation coefficient –

UNIT-5

 Correlation of grouped bivariate data – coefficient of determination Multiple Regression – partial Regression coefficients.

UNIT-6

Tests of significance – Analysis of variance for regression – Multiple correlation coefficients – Multiple linear regression with two independent variables.

UNIT-7

UNIT-8

Exchange method – Greville Algorithm for the Moore-Penrose inverse.  Orthogonal matrix – Gram – Schmidt Orthogonalization process.