**UPSC Mathematics Syllabus**

Download detailed updated **UPSC Mathematics Syllabus**! Entire Mathematics syllabus along with topics as well as sub topics is well listed here. Union Public Service Commission conducts Civil Services Exam for recruiting eligible aspirants for job positions. Those who have chosen Mathematics as an optional subject may refer latest UPSC Maths Syllabus. In Mathematics, questions are asked from topics like Linear Algebra, Calculus, Analytic Geometry, Ordinary Differential Equations, Algebra & others. Access UPSC Mathematics Syllabus 2018 in PDF Form.

Mathematics Syllabus for Paper 1 as well as Paper 2 is well listed here. To score good marks in Mathematics subject, allot time to all topics as listed in syllabus. Practice all important questions & make notes chapter wise. Practice will make your perfect as there are more topics added in paper. To gain more stuff about UPSC Mathematics Syllabus 2018, scroll down this page tailored by team of recruitmentresult.com.

### UPSC Mathematics Syllabus

__UPSC Mathematics Syllabus 2018__:

__UPSC Mathematics Syllabus for Paper 1__:

__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.

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__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.

__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.

__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.

__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.

__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.

Read Now: __How to Prepare for Maths__

__UPSC Mathematics Syllabus for Paper 2__:

__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.

__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.

__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.

__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.

**Get PDF: UPSC Mathematics Syllabus**

__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.

__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

__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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__UPSC Mathematics Preparation Books__

Contenders may refer these books to plan out preparation for UPSC Mathematics Exam.

- R.Vasista Schaum Series Schaum Series (3000 solved problems book)
- C Malik and Savita Arora Shanti Narayana
- N. Chatterjee
- D. Raisinghania
- Ian Sneddon
- R. Vasista
- Joseph A. Gallian
- Shramik Sen Upadhayay
- Schuam Series
- N. Sharma
- Ponnu Swami
- Shanti Swarup
- Kanti Swarup
- Numerical Analysis
- Jain and Iyenger

Union Public Service Commission conducts __Civil Services Examination__ for appointing officers for the All India Administrative civil services. We have listed detailed UPSC Mathematics 2018 Syllabus on this page. Make time table & practice all questions as prescribed in Union Public Service Commission Syllabus. You can also download syllabus in PDF form from this page.

__Final words__:

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