## GATE Examination Syllabus-(XE)Enginering Sciences

**ENGINEERING MATHEMATICS**

**Linear Algebra :** Determinates, algebra of matrices, rank, inverse, system of linear equations, symmetric, skew-symmetric and orthogonal matrices. Hermitian, skew-hermitian and unitary matrices, eigenvalues and eigenvectors, diagonalisation of matrices, Cayley-Hamiltonian, quadratic forms.

**Calculus :** Functions of single variables, limit, continuity and differentiability, Mean value theorems, Intermediate 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, Fourier integrals.

**Complex variable: **Analytic functions, Cauchy’s integral theorem and integral formula without proof. Taylor’s and Laurent’ series, Residue theorem (without proof) with application to the evaluation of real integarls. Vector Calculus: Gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, Stokes, Gauss and Green’s theorems (without proofs) with applications.

Ordinary Differential Equations: First order equation (linear and nonlinear)., higher order linear differential equations with constant coefficients, method of variation of paramaters, Cauchy’s and Euler’s equations, initial and boundary value problems, power series solutions, Legendre polynomials and Bessel’s functions of the first kind. Partial Differential Equations: Variables separable method, solutions of one dimensional heat, wave and Laplace equations.

**Probability and Statistics:** Definitions of probability and simple theorems, conditional probability, mean, mode and standard deviation, random variables, discrete and continuous distributions, Poisson, normal and Binomial distribution, correlation and regression

**Numerical Methods:** L-U decomposition for systems of linear equations,Newton-Raphson method, numerical integration(trapezoidal and Simpson’s rule), numerical methods for first order differential equation (Euler 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, solution of non-linear equations of one variables 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 quadrature 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.

**Programming : **Elementary concepts and terminology of a computer system and system software, Fortran77 and C programming.

Fortran : Program organization, arithmetic statements, transfer of control, Do loops, subscripted variables, functions and subroutines.

**C language : **Basic data types and declarations, flow of control- iterative statement, conditional statement, unconditional branching, arrays, functions and procedures.

**ELECTRICAL SCIENCES**

Electric Circuits: Ideal voltage and current sources; RLC circuits, steady state and transient analysis of DC circuits, network theorems; alternating currents and voltages, single-phase AC circuits, resonance; three-phase circuits.

**Magnetic circuits: **Mmf and flux, and their relationship with voltage and current; transformer, equivalent circuit of a practical transformer, three-phase transformer connections.

**Electrical machines:** Principle of operation, characteristics, efficiency and regulation of DC and synchronous machines; equivalent circuit and performance of three-phase and single-phase induction motors.

**Electronic Circuits: **Characteristics of p-n junction diodes, zener diodes, bipolar junction transistors (BJT) and junction field effect transistors (JFET); MOSFET’s structure, characteristics, and operations; rectifiers, filters, and regulated power supplies; biasing circuits, different configurations of transistor amplifiers, class A, B and C of power amplifiers; linear applications of operational amplifiers; oscillators; tuned and phase “shift types. Digital circuits: Number systems, Boolean algebra; logic gates, combinational circuits, flip-flops (RS, JK, D and T) counters.

**Measuring instruments:** Moving coil, moving iron, and dynamometer type instruments; shunts, instrument transformers, cathode ray oscilloscopes; D/A and A/D converters.

**FLUID MECHANICS**

Fluid Properties: Relation between stress and strain rate for Newtonian fluids

Hydrostatics, buoyancy, manometry

Concept of local and convective accelerations; control volume analysis for mass, momentum and energy conservation. Differential equations of continuity and momentum (Euler’s equation of motion); concept of fluid rotation, stream function, potential function; Bernoulli’s equation and its applications.

Qualitative ideas of boundary layers and its separation; streamlined and bluff bodies; drag and lift forces. Fully-developed pipe flow; laminar and turbulent flows; friction factor; Darcy Weisbach relation; Moody’s friction chart; losses in pipe fittings; flow measurements using venturimeter and orifice plates.

Dimensional analysis; similitude and concept of dynamic similarity; importance of dimensionless numbers in model studies.

**MATERIALS SCIENCE
**

Atomic structure and bonding in materials: metals, ceramics and polymers.

Structure of materials: Crystal systems, unit cells and space lattice; determination of structures of simple crystals.by X-ray diffraction; Miller indices for planes and directions. Packing geometry in metallic, ionic and covalent solids.

Concept of amorphous, single and polycrystalline structures and their effects on properties of materials. Imperfections in crystalline solids and their role in influencing various properties.

Fick’s laws of diffusion and applications of diffusion in sintering, doping of semiconductors and surface hardening of metals.

**Alloys:** solid solution and solubility limit. Binary phase diagram, intermediate phases and intermetallic compounds; iron-iron carbide phase diagram. Phase transformation in steels. Cold and hot working of metals, recovery, recrystallization and grain growth.

Properties and applications of ferrous and nonferrous alloys.

Structure, properties, processing and applications of traditional and advanced ceramics.

**Polymers:** classification, polymerization, structure and properties, additives for polymer products, processing and application.

**Composites:** properties and application of various composites.

Corrosion and environmental degradation of materials (metals, ceramics and polymers).

**Mechanical properties of materials:** Stress-strain diagrams of metallic, ceramic and polymeric materials, modulus of elasticity, yield strength, plastic deformation and toughness, tensile strength and elongation at break; viscoelasticity, hardness, impact strength, ductile and brittle fracture, creep and fatigue properties of materials. Heat capacity, thermal conductivity, thermal expansion of materials.

Concept of energy band diagram for materials; conductors, semiconductors and insulators in terms of energy bands. Electrical conductivity, effect of temperature on conductivity in materials, intrinsic and extrinsic semiconductors, dielectric properties of materials.

Refraction, reflection, absorption and transmission of electromagnetic radiation in solids. Origin of magnetism in metallic and ceramic materials, paramagnetism, diamagnetism, antiferromagnetism, ferromagnetism, ferrimagnetism in materials and magnetic hysteresis. Advanced materials: Smart materials exhibiting ferroelectric, piezoelectric, optoelectronic, semiconducting behaviour; lasers and optical fibers; photoconductivity and superconductivity in materials.

**SOLID MECHANICS**

Equivalent force systems; free-body diagrams; equilibrium equations; analysis of determinate and indeterminate 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; equilibrium equations; compatibility conditions; yield criteria. 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.

**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 vapour 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.

Ideal Gas Mixtures: Dalton’s and Amagat’s laws, calculations of properties, air-water vapour mixtures.

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