**Mathematics Syllabus UPSC Mains **

**Mathematics Syllabus Paper – I**

**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.

**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 integrals; 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.

**Mathematics Syllabus Paper – II**

**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.

**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 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. - 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 **

- 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; 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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