A composite paper covering engineering mathematics (compulsory) and optional sections like fluid mechanics, materials science, and thermodynamics, for interdisciplinary engineering roles.

GATE Engineering Sciences

Official 2025 syllabus extracted from https://gate2025.iitr.ac.in/doc/2025/GATE%20_XE_2025_Syllabus.pdf

XE-A: Engineering Mathematics (Compulsory)

Algebra of real matrices: determinant, inverse and rank

System of linear equations: unique, no, and infinite solutions

Eigen values and eigen vectors

Properties of eigen values/vectors of symmetric matrices

Diagonalization of matrices

Cayley-Hamilton theorem

Linear Algebra

Limit, indeterminate forms, L'Hospital's rule

Continuity and differentiability

Mean value theorems

Maxima and minima

Taylor's theorem

Fundamental theorem and mean value theorem of integral calculus

Definite and improper integrals

Areas and volumes by rotation

Functions of two variables: limit, continuity, partial derivatives

Directional and total derivatives

Maxima, minima, saddle points

Lagrange multipliers

Double integrals

Sequences and series convergence

Tests of convergence (ratio, root, integral)

Power series

Taylor's series

Fourier series of period 2π

Calculus

Gradient, divergence and curl

Line integrals

Green's theorem

Vector Calculus

Complex numbers

Argand plane and polar representation

De Moivre's theorem

Analytic functions

Cauchy-Riemann equations

Complex Variables

First order linear and nonlinear equations

Second order linear ODEs with constant coefficients

Cauchy-Euler equation

Second order linear ODEs with variable coefficients

Wronskian

Method of variation of parameters

Eigen value problem for second order equations

Power series solutions for ordinary points

Ordinary Differential Equations

Classification of second order linear PDEs

Method of separation of variables

1-D heat equation

2-D Laplace equation

Partial Differential Equations

Axioms of probability

Conditional probability

Bayes' theorem

Mean, variance and standard deviation

Binomial, Poisson and Normal distributions

Correlation and linear regression

Probability and Statistics

LU decomposition

Gauss elimination

Lagrange and Newton interpolation

Newton-Raphson method

Trapezoidal and Simpson's rules

Explicit Euler's method for ODEs

Numerical Methods

XE-B: Fluid Mechanics

Density, viscosity, surface tension

Newtonian fluids: stress-strain rate relationship

Viscous vs inviscid, compressible vs incompressible flows

Internal vs external, steady vs unsteady, laminar vs turbulent

1D/2D/3D flows

Newtonian vs non-Newtonian fluid flow

Buoyancy, manometry, forces on submerged bodies and stability

Flow and Fluid Properties

Eulerian and Lagrangian descriptions

Local, convective and material derivatives

Streamline, streakline, pathline, timeline

Kinematics of Fluid Motion

Reynolds Transport Theorem for mass, linear and angular momentum

Integral Analysis for Control Volume

Differential equations of mass and momentum for incompressible flows

Euler equations (inviscid) and Navier-Stokes equations (viscous)

Fluid rotation, vorticity, stream function, circulation

Exact solutions: Couette flow, Poiseuille flow, thin film flow

Differential Analysis

Geometric, kinematic and dynamic similarity

Buckingham Pi theorem

Reynolds, Froude and Mach numbers

Dimensional Analysis

Fully developed pipe flow

Friction factor for laminar and turbulent flows

Darcy-Weisbach relation and Moody's chart

Major and minor losses

Internal Flows

Bernoulli's equation: assumptions and applications

Flow measurements: Venturi meter, Pitot-static tube, orifice meter

Velocity potential function

Uniform flow, source, sink, vortex and their superposition

Bernoulli's Equation and Potential Flows

Prandtl boundary layer equations: concept and assumptions

Boundary layer thickness, displacement and momentum thickness

Boundary layer separation

Streamlined and bluff bodies

Drag and lift forces

External Flows

XE-C: Materials Science

Metals, ceramics, polymers and composites

Metallic, ionic, covalent and mixed bonding

Crystallography fundamentals, symmetry operations, crystal systems

Bravais lattices, unit cells, primitive cells

Crystallographic planes and directions

Structures of metals, ceramics, polymers, amorphous materials and glasses

0-D, 1-D and 2-D defects

Vacancies, interstitials, solid solutions

Frenkel and Schottky defects

Dislocations

Grain boundaries, twins, stacking faults

Surfaces and interfaces

Classification and Structure of Materials

Extensive and intensive thermodynamic properties

Laws of thermodynamics

Phase equilibria, phase rule

Phase diagrams (unary and binary)

Basic electrochemistry

Reaction kinetics

Fundamentals of diffusion, Fick's laws

Solidification of pure metals and alloys

Nucleation and growth

Diffusional solid-state phase transformations: precipitation and eutectoid

Martensitic transformation

Thermodynamics, Kinetics and Phase Transformations

Mechanical properties of metals, ceramics, polymers and composites

Stress-strain response: elastic, anelastic and plastic deformation

Free electron theory, Fermi energy, density of states

Elements of band theory, semiconductors

Hall effect, dielectric behaviour

Piezo- and ferro-electric behaviour

Origin of magnetism: para-, dia-, ferro- and ferri-magnetism

Specific heat, heat conduction, thermal diffusivity, thermal expansion

Thermoelectricity

Refractive index, absorption and transmission of electromagnetic radiation

Properties and Applications of Materials

X-ray diffraction

Spectroscopic techniques: UV-Vis, IR and Raman

Optical and electron microscopy

Composition analysis in electron microscopes

Tensile test, hardness measurement

Electrical conductivity, carrier mobility and concentrations

Thermal analysis: thermogravimetry and calorimetry

Characterization and Measurements of Properties

Heat treatment of ferrous and aluminium alloys

Preparation of ceramic powders, sintering

Thin film deposition: evaporation and sputtering

Chemical vapour deposition

Thin film growth phenomena

Processing of Materials

Corrosion and its prevention

Embrittlement of metals

Polymer degradation

Degradation of Materials

XE-D: Solid Mechanics

Equivalent force systems

Free-body diagrams

Equilibrium equations

Analysis of determinate trusses and frames

Friction

Principle of minimum potential energy

Particle kinematics and dynamics

Dynamics of rigid bodies under planar motion

Law of conservation of energy

Law of conservation of momentum

Mechanics of Rigid Bodies

Stresses and strains

Transformation of 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 force, shear force 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

Mechanics of Deformable Bodies

Free vibration of undamped single degree of freedom systems

Vibrations

XE-E: Thermodynamics

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

Concepts of heat and work, modes of work

Basic Concepts

GATE Engineering Sciences Syllabus 2025, 2026 and more: Subjects, Topics, and What to Study