Table of Contents

- 1 UPSC Maths Syllabus
- 1.1 UPSC Mains Maths Paper Syllabus – Linear Algebra:
- 1.2 UPSC Mathematics Syllabus of Calculus:
- 1.3 UPSC Math Syllabus – Analytic Geometry:
- 1.4 UPSC Mathematics Syllabus of Ordinary Differential Equations:
- 1.5 UPSC Maths Syllabus – Dynamics & Statics:
- 1.6 UPSC Mathematics Syllabus Vector Analysis:
- 1.7 UPSC Maths Paper Syllabus –Algebra:
- 1.8 Share this:

**UPSC Maths Syllabus**

**Paper 1**

### 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 **

**Paper 2**

### 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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