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 GATE Syllabus: Engineering Sciences (XE) Engineering Mathematics (Compulsory): Linear Algebra: Algebra of matrices, inverse, rank, system of linear equations, symmetric, skewsymmetric and orthogonal matrices. Hermitian, skew-Hermitian and unitary matrices. eigenvalues and eigenvectors, diagonalisation of matrices, Cayley-Hamilton Theorem. Calculus: Functions of single variable, limit, continuity and differentiability, Mean value theorems, Indeterminate forms and L'Hospital rule, Maxima and minima, Taylor's series, Fundamental and mean value-theorems of integral calculus. Evaluation of definite and improper integrals, Beta and Gamma functions, Functions of two variables, limit, continuity, partial derivatives, Euler's theorem for homogeneous functions, total derivatives, maxima and minima, Lagrange method of multipliers, double and triple integrals and their applications, sequence and series, tests for convergence, power series, Fourier Series, Half range sine and cosine series. Complex variable: Analytic functions, Cauchy-Riemann equations, Application in solving potential problems, Line integral, Cauchy's integral theorem and integral formula (without proof), Taylor's and Laurent' series, Residue theorem (without proof) and its applications. Vector Calculus: Gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, Stokes, Gauss and Green's theorems (without proofs) applications. Ordinary Differential Equations: First order equation (linear and nonlinear), Second order linear differential equations with variable coefficients, Variation of parameters method, higher order linear differential equations with constant coefficients, Cauchy- Euler's equations, power series solutions, Legendre polynomials and Bessel's functions of the first kind and their properties. Partial Differential Equations: Separation of variables method, Laplace equation, solutions of one dimensional heat and wave equations. Probability and Statistics: Definitions of probability and simple theorems, conditional probability, Bayes Theorem, random variables, discrete and continuous distributions, Binomial, Poisson, and normal distributions, correlation and linear regression. Numerical Methods: Solution of a system of linear equations by L-U decomposition, Gauss- Jordan and Gauss-Seidel Methods, Newton’s interpolation formulae, Solution of a polynomial and a transcendental equation by Newton-Raphson method, numerical integration by trapezoidal rule, Simpson’s rule and Gaussian quadrature, numerical solutions of first order differential equation by Euler’s method and 4th order Runge-Kutta method. Computational Science: Numerical Methods: Truncation errors, round off errors and their propagation; Interpolation: Lagrange, Newton's forward, backward and divided difference formulas, Least square curve fitting; Solutions of non linear equations of one variable using bisection, false position, Secant and Newton Raphson methods, Rate of convergence of these methods, general iterative methods, Simple and multiple roots of polynomials; Solutions of system of linear algebraic equations using Gauss elimination methods, Jacobi and Gauss - Seidel iterative methods and their rate of convergence; Ill conditioned and well conditioned system, Eigen values and Eigen vectors using power methods; Numerical integration using trapezoidal, Simpson's rule and other quadtrature formulas; Numerical Differentiation; Solution of boundary value problems; Solution of initial value problems of ordinary differential equations using Euler's method, predictor corrector and Runge Kutta method. Computer System Concepts: Representation of fixed- and floating-point numbers; Elementary concepts and terminology of basic building blocks of a computer system and system software. Fortran: Fortran-90 for Numerical Computation: Basic data types including complex numbers; Arrays; Assignment statements; Structured Programming Constructs: Loops, Conditional execution, iteration and recursion; Functions and subroutines; Structured programming practices. C language: Basic data types including pointers; Assignments statements; Control statements; Dynamic memory allocation; Functions and procedures; Parameter passing mechanisms; Structured programming practices. Electrical Sciences: Electric Circuits: Ideal voltage and current sources; RLC circuits, steady state and transient analysis of DC circuits, network theorems; single phase AC circuits, resonance and three phase circuits. Magnetic Circuits: MMF and flux, and their relationship with voltage and current; principle of operation of transformer, equivalent circuit of a practical transformer, efficiency and regulation of transformer. Electric Machines: Principle of operation, characteristics and performance equations of DC machines; principle of operation, equivalent circuit of three-phase Induction machine Electronic Circuits: Characteristics of p-n junction diode, Zener diode, bi-polar junction transistor (BJT) and junction field effect transistor (JFET); structure of MOSFET, its characteristics and operation; rectifiers, filters, and regulated power supply, transistor biasing circuits, operational amplifiers, linear applications of operational amplifier, oscillators (tuned and phase shift type) Digital circuits: Number systems, Boolean algebra, logic gates, combinational and sequential circuits, Flip-Flops (RS, JK, D and T), Counters. Measuring Instruments: Cathode Ray oscilloscope, D/A and A/D converters. Fluid Mechanics: Fluid Properties: Relation between stress and strain rate for Newtonian fluids. Hydrostatics: Buoyancy, manometry, forces on submerged bodies. Eulerian and Lagrangian description of fluid motion, concept of local and convective accelerations, steady and unsteady flows, control volume analysis for mass, momentum and energy. Differential equations of mass and momentum (Euler equation), Bernoulli’s equation and its applications. Concept of fluid rotation, vorticity, stream function and potential function. Potential flow: elementary flow fields and principle of superposition, potential flow past a circular cylinder. Dimensional analysis: Concept of geometric, kinematic and dynamic similarity, importance of non-dimensional numbers. Fully-developed pipe flow, laminar and turbulent flows, friction factor, Darcy-Weisbach relation. Qualitative ideas of boundary layer and separation, streamlined and bluff bodies, drag and lift forces. Basic ideas of flow measurement using venturimeter, pitot-static tube and orifice plate. Materials Science: Structure: Atomic structure and bonding in materials. Crystal structure of materials, crystal systems, unit cells and space lattices, determination of structures of simple crystals by x-ray diffraction, miller indices of planes and directions, packing geometry in metallic, ionic and covalent solids. Concept of amorphous, single and polycrystalline structures and their effect on properties of materials. Crystal growth techniques. Imperfections in crystalline solids and their role in influencing various properties. Diffusion: Fick’s laws and application of diffusion in sintering, doping of semiconductors and surface hardening of metals. Metals and Alloys: Solid solutions, solubility limit, phase rule, binary phase diagrams, intermediate phases, intermetallic compounds, iron-iron carbide phase diagram, heat treatment of steels, cold, hot working of metals, recovery, recrystallization and grain growth. Microstrcture, properties and applications of ferrous and non-ferrous alloys. Ceramics: Structure, properties, processing and applications of traditional and advanced ceramics. Polymers: Classification, polymerization, structure and properties, additives for polymer products, processing and applications. Composites: Properties and applications of various composites. Advanced Materials and Tools: Smart materials, exhibiting ferroelectric, piezoelectric, optoelectric, semiconducting behavior, lasers and optical fibers, photoconductivity and superconductivity, nanomaterials – synthesis, properties and applications, biomaterials, superalloys, shape memory alloys. Materials characterization techniques such as, scanning electron microscopy, transmission electron microscopy, atomic force microscopy, scanning tunneling microscopy, atomic absorption spectroscopy, differential scanning calorimetry. Mechanical Properties: stress-strain diagrams of metallic, ceramic and polymeric materials, modulus of elasticity, yield strength, tensile strength, toughness, elongation, plastic deformation, viscoelasticity, hardness, impact strength, creep, fatigue, ductile and brittle fracture. Thermal Properties: Heat capacity, thermal conductivity, thermal expansion of materials. Electronic Properties: Concept of energy band diagram for materials - conductors, semiconductors and insulators, electrical conductivity – effect of temperature on conductility, intrinsic and extrinsic semiconductors, dielectric properties. Optical Properties: Reflection, refraction, absorption and transmission of electromagnetic radiation in solids. Magnetic Properties: Origin of magnetism in metallic and ceramic materials, paramagnetism, diamagnetism, antiferro magnetism, ferromagnetism, ferrimagnetism, magnetic hysterisis. Environmental Degradation: Corrosion and oxidation of materials, prevention. Solid Mechanics: Equivalent force systems; free-body diagrams; equilibrium equations; analysis of determinate trusses and frames; friction; simple relative motion of particles; force as function of position, time and speed; force acting on a body in motion; laws of motion; law of conservation of energy; law of conservation of momentum. Stresses and strains; principal stresses and strains; Mohr's circle; generalized Hooke's Law; thermal strain; theories of failure. Axial, shear and bending moment diagrams; axial, shear and bending stresses; deflection (for symmetric bending); torsion in circular shafts; thin cylinders; energy methods (Castigliano's Theorems); Euler buckling. Free vibration of single degree of freedom systems. Thermodynamics: Basic Concepts: Continuum, macroscopic approach, thermodynamic system (closed and open or control volume); thermodynamic properties and equilibrium; state of a system, state diagram, path and process; different modes of work; Zeroth law of thermodynamics; concept of temperature; heat. First Law of Thermodynamics: Energy, enthalpy, specific heats, first law applied to systems and control volumes, steady and unsteady flow analysis. Second Law of Thermodynamics: Kelvin-Planck and Clausius statements, reversible and irreversible processes, Carnot theorems, thermodynamic temperature scale, Clausius inequality and concept of entropy, principle of increase of entropy; availability and irreversibility. Properties of Pure Substances: Thermodynamic properties of pure substances in solid, liquid and vapor phases, P-V-T behaviour of simple compressible substances, phase rule, thermodynamic property tables and charts, ideal and real gases, equations of state, compressibility chart. Thermodynamic Relations: T-ds relations, Maxwell equations, Joule-Thomson coefficient, coefficient of volume expansion, adiabatic and isothermal compressibilities, Clapeyron equation. Thermodynamic cycles: Carnot vapor power cycle, Ideal Rankine cycle, Rankine Reheat cycle, Air standard Otto cycle, Air standard Diesel cycle, Air-standard Brayton cycle, Vapor-compression refrigeration cycle. Ideal Gas Mixtures: Dalton’s and Amagat’s laws, calculations of properties, air-water vapor mixtures and simple thermodynamic processes involving them.