For a UPSC CSE aspirant, the optional subject is also an important subject. In the UPSC mains exam, optional marks have two papers, Paper 1 and Paper 2. Each paper is of 250 marks which makes a total of 500 marks. The UPSC optional subject list contains 48 subjects in total, one of which is Mathematics.

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**SYLLABUS FOR PAPER I**

**SECTION A **

**1. Linear Algebra:**

- Space, vector, linear independence, and dependence, bases, dimensions, subspaces
- Finite-dimensional vector spaces
- Matrices, Cayley-Hamilton theorem, eigenvectors and eigenvalues, a matrix of a linear transformation, Echelon form, row and column reduction, equivalence, congruences and similarity, rank, orthogonal, reduction to canonical form, symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian forms–their eigenvalues
- Orthogonal and unitary reduction of quadratic and hermitian forms, positive definite quadratic forms.

**2. Calculus: **

- Real numbers, limits, continuity, differentiability, mean-value theorems, Taylor’s theorem with remainders, indeterminate forms, maxima, and minima, asymptotes
- Functions of several variables: continuity, differentiability, partial derivatives, maxima and minima, Jacobian, Lagrange’s method of multipliers
- Areas, surface and volumes, center of gravity
- Riemann’s definition of definite integrals, indefinite integrals, infinite and improper integrals, beta, and gamma functions
- Double and triple integrals (evaluation techniques only)

**3. Analytic Geometry:**

- Cartesian and polar coordinates in two and three dimensions, reduction to canonical forms, straight lines, second-degree equations in two and three dimensions, the shortest distance between two skew lines, plane, sphere, cone, cylinder., paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

**Section-B **

**1. Ordinary Differential Equations: **

- Formulation of differential equations, order and degree, equations of the first order and first degree, integrating factor, equations of first order but not of the first degree, Clariaut’s equation, singular solution
- Higher-order linear equations, with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation
- Second-order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.

**2. Dynamics, Statics and Hydrostatics:**

- Degree of freedom and constraints, simple harmonic motion, motion in a plane, projectiles, constrained rectilinear motion, motion, motion under impulsive forces, work, and energy, conservation of energy, Kepler’s laws, orbits under central forces, the motion of the varying mass, motion under resistance
- Equilibrium of a system of particles, work, and potential energy, friction, common catenary, the equilibrium of forces in three dimensions, the stability of equilibrium, the principle of virtual work
- Bernoulli’s equation, the pressure of heavy fluids, the equilibrium of fluids under a given system of forces, the center of pressure, thrust on curved surfaces, the stability of equilibrium, the equilibrium of floating bodies, metacenter, the pressure of gases.

**3. Vector Analysis:**

- Scalar and vector fields, triple, products, differentiation of vector function of a scalar variable, divergence, gradient and curl in cartesian, cylindrical, and spherical coordinates and their physical interpretations
- Application to Geometry: Curves in space, curvature, and torsion. Serret-Frenet’s formulae, Gauss and Stokes’ theorems, Green’s identities.
- Higher-order derivatives, vector equations, and identities

### SYLLABUS FOR PAPER II

**Section-A **

**1. Algebra:**

- Groups, homomorphism of groups, subgroups, normal subgroups, quotient groups basic isomorphism theorems, Sylow’s group, permutation groups, Cayley theorem
- Field extensions, finite fields
- Rings and ideals, principal ideal domains, unique factorization domains, and Euclidean domains

**2. Real Analysis: **

- Real number system, real number system as an ordered field with least upper bound property, ordered sets, bounds, ordered field, Cauchy sequence, completeness, Continuity and uniform continuity of functions, properties of continuous functions on compact sets
- Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series
- Uniform convergence, continuity, differentiability, and integrability for sequences and series of functions
- Multiple integrals
- Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima, and minima

**3. Complex Analysis:**

- Analytic function, Cauchy’s theorem, Cauchy-Riemann equations, Cauchy’s integral formula, Taylor’s series, power series, Laurent’s Series, Singularities, Cauchy’s residue theorem, contour integration
- Conformal mapping, bilinear transformations

**4. Linear Programming: **

- Linear programming problems, basic solution, basic feasible solution, and optimal solution, graphical method, and simplex method of solutions
- Transportation and assignment problems
- Duality
- Traveling salesman problems

**Section-B **

**1. Partial differential equations:**

- Formulation of partial differential equations, solutions of equations of type dx/p=dy/q=dz/r
- Curves and surfaces in three dimensions
- Orthogonal trajectories, pfaffian differential equations
- Partial differential equations of the first order, solution by Cauchy’s method of characteristics
- Laplace equation
- Charpit’s method of solutions, linear partial differential equations of the second order with constant coefficients, equations of a vibrating string, heat equation

**2. Numerical Analysis and Computer programming: **

- Newton’s (Forward and backward) and Lagrange’s method of interpolation
- Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi, and Newton-Raphson methods, solution of a system of linear equations by Gauss-Jordan (direct) methods and Gaussian elimination, Gauss-Seidel(iterative) method

**3. Numerical integration:**

- Simpson’s one-third rule, Gaussian quadrature formula, trapezoidal rule

**4. Numerical solution of ordinary differential equations:**

- Euler and Runge Kutta-methods
- Computer Programming: Storage of numbers in Computers, bits, bytes and words, binary system. arithmetic and logical operations on numbers
- Bitwise operations
- Octal and Hexadecimal Systems
- AND, OR, XOR, NOT, and shift/rotate operators
- Conversion to and from decimal Systems
- Algorithms and flow charts for solving numerical analysis problems
- Representation of unsigned integers, signed integers and reals, double precision reals, and long integers
- Developing simple programs in Basic for problems involving techniques covered in the numerical analysis

**5. Mechanics and Fluid Dynamics:**

- Generalized coordinates, constraints, holonomic and non-holonomic, systems
- D’ Alembert’s principle and Lagrange’s equations, Hamilton equations, the moment of inertia, the motion of rigid bodies in two dimensions
- Navier-Stokes equation for a viscous fluid
- Equation of continuity, Euler’s equation of motion for inviscid flow, stream-lines, the path of a particle, potential flow, two-dimensional and axisymmetric motion, sources and sinks, vortex motion, flow past a cylinder and a sphere, method of images