## GATE Syllabus for Engineering Sciences (XE)

Please note – The GATE exam will also have General Aptitude section. The **General Aptitude section GATE syllabus** is published separately.

**Section A: Engineering Mathematics (Compulsory)**

**Linear Algebra:** Algebra of matrices; Inverse and rank of a matrix; System of linear equations; Symmetric, skew-symmetric and orthogonal matrices; Determinants; 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’s rule; Maxima and minima; Taylor’s theorem; Fundamental theorem and mean value-theorems of integral calculus; Evaluation of definite and improper integrals; Applications of definite integrals to evaluate areas and volumes.

Functions of two variables: Limit, continuity and partial derivatives; Directional derivative; Total derivative; Tangent plane and normal line; Maxima, minima and saddle points; Method of Lagrange multipliers; Double and triple integrals, and their applications.

Sequence and series: Convergence of sequence and series; Tests for convergence; Power series; Taylor’s series; Fourier Series; Half range sine and cosine series.

**Vector Calculus:** Gradient, divergence and curl; Line and surface integrals; Green’s theorem, Stokes theorem and Gauss divergence theorem (without proofs).

**Complex variable:** Analytic functions; Cauchy-Riemann equations,Line integral, Cauchy’s integral theorem and integral formula (without proof), Taylor’s series and Laurent series,Residue theorem (without proof) and its 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:** Classification of second order linear partial differential equations,Method of separation of variables, Laplace equation, Solutions of one dimensional heat and wave equations

**Probability and Statistics**: Axioms of probability, Conditional probability, Bayes’ Theorem, Discrete and continuous random variables: Binomial, Poisson and normal distributions, Correlation and linear

regression.

**Numerical Methods:** Solution of systems of linear equations using LU decomposition, Gauss elimination and Gauss-Seidel methods, Lagrange and Newton’s interpolations, Solution of polynomial and transcendental equations by Newton-Raphson method, Numerical integration by trapezoidal rule, Simpson’s rule and Gaussian quadrature rule, Numerical solutions of first order differential equations by Euler’s method and 4th order Runge-Kutta method.

**Section B: Fluid Mechanics**

** Flow and Fluid :** viscosity, relationship between stress and strain-rate for Newtonian fluids, incompressible and compressible flows, differences between laminar and turbulent flows. Hydrostatics: Buoyancy, manometry, forces on submerged bodies.

**Kinematics:** Eulerian and Lagrangian description of fluids motion, concept of local and convective accelerations, steady and unsteady flows.

**Integral analysis:** Control volume analysis for mass, momentum and energy

**Differential Analysis:** Differential equations of mass and momentum for incompressible flows: inviscid – Euler equation and viscous flows – Navier-Stokes equations, concept of fluid rotation, vorticity, stream function, Exact solutions of Navier-Stokes equation for Couette Flow and Poiseuille flow.

**Inviscid flows : **Bernoulli’s equation – assumptions and applications, potential function, Elementary plane flows – uniform flow, source, sink and doublet and their superposition for potential flow past simple geometries.

**Dimensional analysis :** Concept of geometric, kinematic and dynamic similarity, some common non-dimensional parameters and their physical significance: Reynolds number, Froude number and Mach number.

** Internal flows :** Fully developed pipe flow, empirical relations for laminar and turbulent flows: friction factor and Darcy-Weisbach relation.

**Prandtl boundary layer equations :** Concept and assumptions, qualitative idea of boundary layer and separation, streamlined and bluff bodies, drag and lift forces. Flow measurements: Basic ideas of flow measurement using venturimeter, pitot-static tube and orifice plate.

**Section C: Materials Science**

**Processing of Materials:** Powder synthesis, sintering, chemical methods, crystal growth techniques, zone refining, preparation of nanoparticles and thin films

**Characterisation Techniques:** X-ray diffraction, spectroscopic techniques like UV-vis, IR, Raman. Optical and Electron microscopy

**Structure and Imperfections:** Crystal symmetry, point groups, space groups, indices of planes, close packing in solids, bonding in materials, coordination and radius ratio concepts, point defects, dislocations, grain boundaries, surface energy and equilibrium shapes of crystals

**Thermodynamics and Kinetics:** Phase rule, phase diagrams, solid solutions, invariant reactions, lever rule, basic heat treatment of metals, solidification and phase transformations, Fick’s laws of diffusion, mechanisms of diffusion, temperature dependence of diffusivity

**Properties of Materials:**

**Mechanical Properties:**Stress-strain response of metallic, ceramic and polymer materials, yield strength, tensile strength and modulus of elasticity, toughness, plastic deformation, fatigue, creep and fracture**Electronic Properties:**Free electron theory, Fermi energy, density of states, elements of band theory, semiconductors, Hall effect, dielectric behaviour, piezo, ferro, pyroelectricmaterials**Magnetic Properties:**Origin of magnetism in metallic and ceramic materials, paramagnetism, diamagnetism, ferro and ferrimagnetism**Thermal Properties:**Specific heat, thermal conductivity and thermal expansion, thermoelectricity**Optical Properties:**Refractive index, absorption and transmission of electromagnetic radiation in solids, electrooptic and magnetoopticmaterials, spontaneous and stimulated emission, gas and solid state lasers

**Material types :** Concept of amorphous, single crystals and polycrystalline materials, crystallinity and its effect on physical properties, metal, ceramic, polymers, classification of polymers, polymerization, structure and properties, additives for polymer products, processing and applications, effect of environment on materials, composites

** Environmental Degradation :** Corrosion, oxidation and prevention

** Elements of Quantum Mechanics and Mathematics :** Basics of quantum mechanics, quantum mechanical treatment of electrical, optical and thermal properties of materials, analytical solid geometry, differentiation and integration, differential equations, vectors and tensors, matrices, Fourier series, complex analysis, probability and statistics

**Section D: Solid Mechanics**

Equivalent force systems, free-body diagrams, equilibrium equations, analysis of determinate trusses and frames, friction, particle kinematics and dynamics, dynamics of rigid bodies under planar motion, law of conservation of energy, law of conservation of momentum.

Stresses and strains, principal stresses and strains, Mohr’s circle for plane stress and plane strain, generalized Hooke’s Law, elastic constants, thermal stresses, theories of failure.

Axial, shear and bending moment diagrams, axial, shear and bending stresses, combined stresses, deflection (for symmetric bending), torsion in circular shafts, thin walled pressure vessels, energy methods (Castigliano’s Theorems), Euler buckling.

Free vibration of single degree of freedom systems.

**Section E: Thermodynamics**

**Basic Concepts:** Continuum and macroscopic approach; thermodynamic systems (closed and open); thermodynamic properties and equilibrium; state of a system, state postulate for simple compressible substances, state diagrams, paths and processes on state diagrams; concepts of heat and work, different modes of work; zeroth law of thermodynamics; concept of temperature.

**First Law of Thermodynamics:** Concept of energy and various forms of energy; internal energy, enthalpy; specific heats; first law applied to elementary processes, closed systems and control volumes, steady and unsteady flow analysis.

**Second Law of Thermodynamics:** Limitations of the first law of thermodynamics, concepts of heat engines and heat pumps/refrigerators, Kelvin-Planck and Clausius statements and their equivalence; reversible and irreversible processes; Carnot cycle and Carnot principles/theorems; thermodynamic temperature scale; Clausius inequality and concept of entropy; microscopic interpretation of entropy, the principle of increase of entropy, T-s diagrams; second law analysis of control volume; availability and irreversibility; third law of thermodynamics.

**Properties of Pure Substances:** Thermodynamic properties of pure substances in solid, liquid and vapor phases; P-vT behaviour of simple compressible substances, phase rule, thermodynamic property tables and charts, ideal and real gases, ideal gas equation of state and van der Waals equation of state; law of corresponding states, compressibility factor and generalized compressibility chart.

**Thermodynamic Relations:** T-ds relations, Helmholtz and Gibbs functions, Gibbs relations, Maxwell relations, Joule-Thomson coefficient, coefficient of volume expansion, adiabatic and isothermal compressibilities, Clapeyron and Clapeyron-Clausius equations.

**Thermodynamic cycles:** Carnot vapor 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, properties of ideal gas mixtures, air-water vapor mixtures and simple thermodynamic processes involving them; specific and relative humidities, dew point and wet bulb temperature, adiabatic saturation temperature, psychrometric chart.

**Section F: Polymer Science and Engineering**

**Chemistry of high polymers:** Monomers, functionality, degree of polymerizations, classification of polymers, glass transition, melting transition, criteria for rubberiness, polymerization methods:

addition and condensation, their kinetics, metallocene polymers and other newer techniques of polymerization, copolymerization, monomer reactivity ratios and its significance, kinetics, different copolymers, random, alternating, azeotropic copolymerization, block and graft copolymers, techniques for copolymerizationbulk, solution, suspension, emulsion.

**Polymer Characterization:** Solubility and swelling, concept of average molecular weight, determination of number average, weight average, viscosity average and Z-average molecular weights, polymer crystallinity, analysis of polymers using IR, XRD, thermal (DSC, DMTA, TGA), microscopic (optical and electronic) techniques.

**Synthesis and properties:** Commodity and general purpose thermoplastics: PE, PP, PS, PVC, Polyesters, Acrylic, PU polymers. Engineering Plastics: Nylon, PC, PBT, PSU, PPO, ABS, Fluoropolymers Thermosetting polymers: PF, MF, UF, Epoxy, Unsaturated polyester, Alkyds. Natural and synthetic rubbers: Recovery of NR hydrocarbon from latex, SBR, Nitrile, CR, CSM, EPDM, IIR, BR, Silicone, TPE.

**Polymer blends and composites:** Difference between blends and composites, their significance, choice of polymers for blending, blend miscibility-miscible and immiscible blends, thermodynamics,

phase morphology, polymer alloys, polymer eutectics, plastic-plastic, rubber-plastic and rubber-rubber blends, FRP, particulate, long and short fibre reinforced composites.

**Polymer Technology:** Polymer compounding-need and significance, different compounding ingredients for rubber and plastics, cross-linking and vulcanization, vulcanization kinetics.

**Polymer rheology:** Flow of Newtonian and non-Newtonian fluids, different flow equations,

dependence of shear modulus on temperature, molecular/segmental deformations at different zones and transitions. Measurements of rheological parameters by capillary rotating, parallel plate, cone-plate rheometer. Viscoelasticity-creep and stress relaxations, mechanical models, control of rheological

characteristics through compounding, rubber curing in parallel plate viscometer, ODR and MDR.

**Polymer processing:** Compression molding, transfer molding, injection molding, blow molding, reaction injection molding, extrusion, pultrusion, calendaring, rotational molding, thermoforming, rubber processing in two-roll mill, internal mixer.

**Polymer testing:** Mechanical-static and dynamic tensile, flexural, compressive, abrasion, endurance, fatigue, hardness, tear, resilience, impact, toughness. Conductivity-thermal and electrical, dielectric constant, dissipation factor, power factor, electric resistance, surface resistivity, volume resistivity, swelling, ageing resistance, environmental stress cracking resistance.

**Section G: Food Technology**

**Food Chemistry and Nutrition:** Carbohydrates: structure and functional properties of mono-, oligo-, & polysaccharides including starch, cellulose, pectic substances and dietary fibre, gelatinization and retrogradation of starch. Proteins: classification and structure of proteins in food, biochemical changes in post mortem and tenderization of muscles. Lipids: classification and structure of lipids, rancidity, polymerization and polymorphism. Pigments: carotenoids, chlorophylls, anthocyanins, tannins and myoglobin. Food flavours: terpenes, esters, aldehydes, ketones and quinines. Enzymes: specificity, simple and inhibition kinetics, coenzymes, enzymatic and nonenzymatic browning. Nutrition: balanced diet, essential amino acids and essential fatty acids, protein efficiency ratio, water soluble and fat soluble vitamins, role of minerals in nutrition, co-factors, anti-nutrients, nutraceuticals, nutrient deficiency diseases. Chemical and biochemical c hanges: changes occur in foods during different processing.

**Food Microbiology:** Characteristics of microorganisms: morphology of bacteria, yeast, mold and actinomycetes, spores and vegetative cells, gram-staining. Microbial growt h: growth and death kinetics, serial dilution technique. Food spoilage: spoilage microorganisms in different food products including milk, fish, meat, egg, cereals and their products. Toxins from microbes: pathogens and non-pathogens including Staphylococcus, Salmonella, Shigella, Escherichia, Bacillus, Clostridium, and Aspergillus genera. Fermented foods and beverages: curd, yoghurt, cheese, pickles, soya-sauce, sauerkraut, idli, dosa, vinegar, alcoholic beverages and sausage.

**Food Products Technology:** Processing principles: thermal processing, chilling, freezing, dehydration, addition of preservatives and food additives, irradiation, fermentation, hurdle technology, intermediate moisture foods. Food pack aging and storage: packaging materials, aseptic packaging, controlled and modified atmosphere storage. Cereal processing and products: milling of rice, wheat, and maize, parboiling of paddy, bread, biscuits, extruded products and ready to eat breakfast cereals. Oil processing: expelling, solvent extraction, refining and hydrogenation. Fruits a nd vegetables p rocessing: extraction, clarification, concentration and packaging of fruit juice, jam, jelly, marmalade, squash, candies, tomato sauce, ketchup, and puree, potato chips, pickles. Plantation crops processing and products: tea, coffee, cocoa, spice, extraction of essential oils and oleoresins from spices. Milk and milk products processing: pasteurization and sterilization, cream, butter, ghee, icecream, cheese and milk powder. Processing of animal products: drying, canning, and freezing of fish and meat; production of egg powder. Waste utilization: pectin from fruit wastes, uses of by-products from rice milling. Food standards and qu ality maintenance: FPO, PFA, Agmark, ISI, HACCP, food plant sanitation and cleaning in place (CIP).

**Food Engineering:** Mass and energy balance; Momentum transfer: Flow rate and pressure drop relationships for Newtonian fluids flowing through pipe, Reynolds number. Heat transfer: heat transfer by conduction, convection, radiation, heat exchangers. Mass transfer: molecular diffusion and Fick’s law, conduction and convective mass transfer, permeability through single and multilayer films. Mechanical operations: size reduction of solids, high pressure homogenization, filtration, centrifugation, settling, sieving, mixing & agitation of liquid. Thermal operations: thermal sterilization, evaporation of liquid foods, hot air drying of solids, spray and freeze-drying, freezing and crystallization. Mass transfer oper ations: psychrometry, humidification and dehumidification operations.

**Section H: Atmospheric & Ocean Science **

Atmospheric Science : Fundamental of Meteorology, Thermal structure of the atmosphere and its composition, Radiation Balance and Laws, Wind Belts, Monsoon, Climate. Atmospheric Thermodynamics. Hydrostatic equilibrium and: Hydrostatic equation, variation of pressure with height, geopotential, Tropical convection. Atmospheric Electricity. Cloud Physics. Observation Techniques of the Atmosepheric Properties. Fundamental equations. Pressure, gravity, centripetal and Corolis forces, continuity equation in Cartesian and isobaric coordinates, Scale analysis, inertial flow, geostrophic and gradient winds, thermal wind, vorticity. Atmospheric turbulence, baroclinic instabiltiy. Atmosphreric Waves. Tropical meteorology: Trade wind inversion, ITCZ; monsoon trough tropical cyclones, their structure and development theory; monsoon depressions; Climate variability and forcings; Madden-Julian oscillation(MJO), ENSO, QBO (quasi-biennial oscillation) and sunspot cycles. Primitive equations of Numerical Weather Prediction. General Circulation and Climate Modelling. Synoptic weather forecasting, prediction of weather elements such as rain, maximum and minimum temperature and fog. Data Assimilation.

**Ocean Sciences :** Seawater Properties, T-S diagrams, Ocean Observations, Ocean Tide and Waves and their properties. Coastal processes and Estuary Dynamics. coastal zone management. Wind Driven Circulation: Ekman, Sverdrup, Stommel and Munk theories, Inertial currents; geostrophic motion; barotropic and baroclinic conditions; Oceanic eddies. Global conveyor belt circulation. Subtropical gyres; Western boundary currents; equatorial current systems; Current System in the Indian Ocean. Momentum equation, mass conservation, vorticity. Ocean and Wave Modeling, Ocean State Forecasting. Data Assimilation. Ocean Turbulence. Chemical Property of seawater, major and minor elements, their behavior and chemical exchanges across interfaces and residence times in seawater, Element chemistry in atypical conditions-estuaries, Biochemical cycling of nutrients, trace metals and organic matter. Air-sea exchange of important biogenic dissolved gases; carbon dioxide-carbonate system; alkalinity and control of pH; biological pump. Marine Pollution. Primary and secondary production; factors controlling phytoplankton and zooplankton abundance and diversity; nekton and fisheries oceanography.

**Download GATE Syllabus for XE**

- XE-A: Engineering Mathematics
- XE-B: Fluid Mechanics
- XE-C: Materials Science
- XE-D: Solid Mechanics
- XE-E: Thermodynamics
- XE-F: Polymer Science and Engineering
- XE-G: Food Technology
- XE-H: Atmospheric & Ocean Science