# UPSC Mathematics Syllabus Math Optional Subject (IAS Mains)

## UPSC Maths Syllabus

### UPSC Mains Maths Paper Syllabus – Linear Algebra:

• Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension
• Linear transformations, rank and nullity, matrix of a linear transformation, Algebra of Matrices
• Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix
• Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial
• Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

### UPSC Mathematics Syllabus of Calculus:

• Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem
• Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; Curve tracing
• Functions of two or three variables: limits, continuity, partial derivatives, maxima and minima
• Lagrange’s method of multipliers, Jacobian
• Riemann’s definition of definite integrals
• Indefinite integral
• Infinite and improper integrals
• Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

### UPSC Math Syllabus – Analytic Geometry:

• Cartesian And Polar Coordinates In Three Dimensions, Second Degree Equations In Three Variables, Reduction To Canonical Forms
• Straight Lines
• Shortest Distance Between Two Skew Lines
• Plane, Sphere
• Cone
• Cylinder, Paraboloid,
• Ellipsoid, Hyperboloid Of One And Two Sheets And Their Properties.

### UPSC Mathematics Syllabus of Ordinary Differential Equations:

• Formulation of differential equations; Equations of first order and first degree, integrating factor
• Orthogonal trajectory; Equations of first order but not of first degree, Clairaut’s equation, singular solution
• Second and higher order linear equations with constant coefficients, complementary function particular integral and general solution.
Second order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.
Laplace and Inverse Laplace transforms and their properties; Laplace transforms of elementary functions
• Application to initial value problems for 2nd order linear equations with constant coefficients.

### UPSC Maths Syllabus – Dynamics & Statics:

• Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion
• Work and energy, conservation of energy
• Kepler’s laws, orbits under central forces.
Equilibrium of a system of particles; Work and potential energy, friction; common catenary
• Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

### UPSC Mathematics Syllabus Vector Analysis:

• Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in cartesian and cylindrical coordinates
• Higher order derivatives
• Vector identities and vector equations.
• Application to geometry
• Curves in space, Curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

UPSC Maths Syllabus for

### UPSC Maths Paper Syllabus –Algebra:

• Groups, Subgroups, Cyclic Groups, Cosets, Lagrange’s Theorem, Normal Subgroups, Quotient Groups
• Homomorphism Of Groups, Basic Isomorphism Theorems, Permutation Groups, Cayley’s Theorem.
• Rings, Subrings And Ideals
• Homomorphisms Of Rings
• Integral Domains, Principal Ideal Domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

UPSC Math Syllabus of Real Analysis:

• Real Number System As An Ordered Field With Least Upper Bound Property; Sequences, Limit Of A Sequence, Cauchy Sequence,
• Completeness Of Real Line; Series And Its Convergence, Absolute And Conditional Convergence Of Series Of Real And Complex Terms, Rearrangement Of Series.
• Continuity And Uniform Continuity Of Functions, Properties Of Continuous Functions On Compact Sets.
• Riemann Integral, Improper Integrals; Fundamental Theorems Of Integral Calculus.
• Uniform Convergence, Continuity, Differentiability And Integrability For Sequences And Series Of Functions
• Partial Derivatives Of Functions Of Several (Two Or Three) Variables, Maxima And Minima.

UPSC Mathematics Syllabus of Complex Analysis:

• Analytic Functions, Cauchy-Riemann Equations, Cauchy’s Theorem, Cauchy’s Integral Formula,
• Power Series Representation Of An Analytic Function, Taylor’s Series; Singularities; Laurent’s Series
• Cauchy’s Residue Theorem; Contour Integration.

UPSC Mathematics Syllabus of Linear Programming:

• Linear Programming Problems, Basic Solution, Basic Feasible Solution And Optimal Solution; Graphical Method And Simplex Method Of Solutions; Duality
• Transportation And Assignment Problems.

IAS Mains Mathematics Syllabus of Partial differential Equations:

• Family Of Surfaces In Three Dimensions And Formulation Of Partial Differential Equations
• Solution Of Quasilinear Partial Differential Equations Of The First Order, Cauchy’s Method Of Characteristics;Linear Partial Differential Equations Of The Second Order With Constant Coefficients, Canonical Form
• Equation Of A Vibrating String, Heat Equation, Laplace Equation And Their Solutions.

UPSC Math Syllabus for Numerical Analysis and Computer programming:

• Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods; solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation, Lagrange’s interpolation.
• Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula.
• Numerical solution of ordinary differential equations: Euler and Runga Kutta-methods.
• Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary numbers.
• Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.
• Representation of unsigned integers, signed integers and reals, double precision reals and long integers.
• Algorithms and flow charts for solving numerical analysis problems

UPSC Mathematics Syllabus of Mechanics and Fluid Dynamics:

• Generalized Coordinates; D’ Alembert’s Principle And Lagrange’s Equations; Hamilton Equations; Moment Of Inertia
• Motion Of Rigid Bodies In Two Dimensions
• Equation Of Continuity; Euler’s Equation Of Motion For Inviscid Flow; Stream-Lines, Path Of A Particle; Potential Flow
• Two-Dimensional And Axisymmetric Motion; Sources And Sinks, Vortex Motion; Navier-Stokes Equation For A Viscous Fluid.

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