**UPSC Mathematics Syllabus**

To appear in the IAS Prelims / IAS Mains exam for Optional Subject, one must practice through the **UPSC Mathematics Syllabus**. With respect to UPSC Maths Syllabus, visitors can also check Exam Pattern to know the structure of examination.

__UPSC IAS Mains Exam Details__

Union Public Service Commission is all set to conduct UPSC IAS Mains Exam on 01-10-2018. Applicants, who’ve qualified in UPSC CSE Prelims Exam and going to appear in CSE Mains Exam, can start their preparation according to updated UPSC Mathematics Syllabus 2018 PDF.

__Need of UPSC Maths Syllabus 2018-19__

In IAS/ CSE or any others examination, conducted by Union Public Service Commission mathematics plays an important role and you may secure highest marks in this subject. UPSC Mathematics Optional Syllabus helps you to know about the important topics.

Apart from it, you may Practice all important questions & make chapter wise notes also. Practice will make you perfect, so what are you waiting for? Gain more stuff about UPSC Mathematics Syllabus 2018-19 by going through this page of recruitmentresult.com.

**UPSC Mathematics Syllabus 2018**

__UPSC Maths Syllabus For Paper 1__

UPSC Linear Algebra Syllabus:

- 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.

Check Now: __Latest UPSC Syllabus__

__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 2018 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 Maths Syllabus for Paper 2__

UPSC Algebra Syllabus:

- 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 2018 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.

**Get PDF: ****UPSC Mathematics Syllabus PDF**

__UPSC Maths 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.

Check Also: __UPSC Notification__

__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 UPSC Math Syllabus in PDF form from this page.

__Final words__:

Hope, you get detailed UPSC Mathematics Syllabus 2018-19. Follow its Official Link to get more details about UPSC Math Syllabus.

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