__
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.