Candidates with background in engineering, technology and related fields can certainly pick the Mathematics as their optional subject in UPSC exam. It will be much easier for such candidates to prepare for this subject and score good marks in this highly competitive exam. However, for exam preparation, they first need to get complete knowledge of its syllabus. In this article, the interested candidates can find the detailed information related to **UPSC Mathematics syllabus 2023** and study for their exam accordingly.

**UPSC Syllabus for Mathematics**

The syllabus of Mathematics is broadly divided into two papers, namely Paper I and Paper II. There are a total of six different topics covered under Paper I and seven topics under the Paper II. Check below the complete details of the UPSC Mathematics syllabus 2023 from this post below.

**Syllabus for Paper I**

**Linear Algebra:**

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

**Calculus:**

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

**Analytic Geometry:**

Second degree equations in three variables, Cartesian and polar coordinates in three dimensions, 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:**

Second and higher order liner equations with constant coefficients, complementary function, particular integral and general solution, Formulation of differential equations, Equations of first order and first degree, integrating factor, 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, orthogonal trajectory, Equations of first order but not of first degree, Clairaut’s equation, singular solution, Section order linear equations with variable coefficients, Euler-Cauchy equation, Determination of complete solution when one solution is known using method of variation of parameters

**Dynamics and Statics:**

Work and energy, conservation of energy, Kepler’s laws, Rectilinear motion, simple harmonic motion, motion in a plane, projectiles, Constrained motion, orbits under central forces. Equilibrium of a system of particles, Principle of virtual work, Stability of equilibrium, equilibrium of forces in three dimensions, friction, Common catenaries

**Vector Analysis:**

Higher order derivatives, Vector identities and vector equation, Scalar and vector fields, differentiation of vector field of a scalar variable, Gradient, divergence and curl in Cartesian and cylindrical coordinates, Application to geometry: Curves in space, curvature and torsion, Serret-Furenet’s formulae, Gauss and Stokes’ theorems, Green’s identities

**Syllabus for Paper II**

**Algebra:**

Rings, subrings and ideals, homomorphisms of rings, Integral domains, principal ideal domains, Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem, Euclidean domains and unique factorization domains, Fields, quotient fields

**Real Analysis:**

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, Real number system as an ordered field with least upper bound property, Sequences, limit of a sequence, Continuity and uniform continuity of functions, properties of continuous functions on compact sets, Riemann integral, improper integrals, Fundamental theorems of integral calculus, Cauchy sequence, completeness of real line, Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series

**Complex Analysis:**

Cauchy’s theorem, Cauchy’s integral formula, Analytic function, Cauchy-Riemann equations, Cauchy’s residue theorem, power series, representation of an analytic function, Taylor’s series, Singularities, Laurent’s series, 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

**Partial Differential Equations:**

Linear partial differential equations of the second order with constant coefficients, canonical form, 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, Equation of a vibrating string, heat equation, Laplace equation and their solutions

**Numerical Analysis and Computer Programming:**

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 real, double precision real and long integers, Algorithms and flow charts for solving numerical analysis problems, 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-Jorden (direct), Gauss-Seidel (iterative) methods, Newton’s (forward and backward) and interpolation, Lagrange’s interpolation, Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula, Numerical solution of ordinary differential equations: Eular 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

**Mechanics and Fluid Dynamics:**

Euler’s equation of motion for inviscid flow, Stream-lines, path of a particle, Generalised coordinates, D’Alembert’s principle and Lagrange’s equations, Hamilton equations, Moment of inertia, Motion of rigid bodies in two dimensions, Equation of continuity, Potential flow, Two-dimensional and axisymmetric motion, Sources and sinks, vortex motion, Navier-Stokes equation for a viscous fluid

Mathematics**UPSC Mathematics Syllabus: Download PDF**

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