**Welcome to the page of Mathematics Optional Syllabus for UPSC Mains 2018!**

UPSC Mains exam is a second stage of civil services exam and ask subjective questions. IAS aspirants who have filled form with Mathematics optional can find syllabus below.

## UPSC Mains Mathematics Optional Syllabus 2018

**UPSC Maths Optional Paper has two papers. Full syllabus is divided in 13 parts.**

UPSC Maths Optional Paper-I | UPSC Maths Optional Paper-II |

Linear Algebra: | Algebra: |

Calculus: | Real Analysis |

Analytic Geometry | Complex Analysis: |

Ordinary Differential Equations: | Linear Programming: |

Dynamics & Statics: | Partial differential equations: |

Vector Analysis: | Numerical Analysis and Computer programming: |

Mechanics and Fluid Dynamics: |

### UPSC Mathematics Optional Syllabus of Paper – I

| |

UPSC Online Form 2018 | Click Here |

UPSC Prelims Syllabus | Click Here |

UPSC Mains Paper of Common Subjects | Click Here |

Syllabus of Other Optional Papers | Click Here |

#### (1) Linear Algebra:

Vector spaces or linear space

Real vector space and *Complex vector space i.e. *over R and C,

Subspaces, basis, dimension;

Linear dependence and independence,

Rank and nullity of linear transformations,

Linear Transformations and Matrices.

Elementary row and column operations.

Algebra of Matrices.

Columns & Row reduction (of a matrix to its row echelon form) congruence’s and similarity;

Notion of Rank of a matrix;

Inverse of a matrix;

Introduction of system of linear equations;

Eigenvalues and eigenvectors of Matrix,

Eigenvalues and characteristic polynomial,

Cayley-Hamilton theorem, Hermitian, skew-Hermitian, Symmetric, skew-symmetric, orthogonal.

Unitary matrices and their eigenvalues.

#### (2) Calculus:

Real numbers, functions of a real variable,

Limits and continuity,

Basis & rules for differentiability,

Taylor’s theorem with remainders,

Mean value theorem,

Indeterminate forms,

Maxima and minima, asymptotes;

Curve tracing;

Functions of two or more variables: Partial derivatives, Lagrange multipliers and Jacobean rank.

Riemann’s definite integrals; Indefinite integrals; Infinite or integrand and improper integrals;

Evaluation Techniques of Double and triple integrals. (Areas, surface and volumes)

#### (3) Analytic Geometry:

Cartesian and polar coordinates in three dimensions,

Second degree equations in three variables,

Reduction to canonical forms,

Shortest distance between two skew lines;

Straight lines, Plane, cone, sphere, and cylinder.

Paraboloid, hyperboloid, ellipsoid of one and two sheets and their properties.

#### (4) Ordinary Differential Equations:

Formulation of differential equations;

First order and first degree equations,

Orthogonal trajectory;

Equations of first order but not of first degree,

Clairaut’s equation, singular solution.

Higher order linear equations with constant coefficients,

Complementary function, particular integral and general solution.

Second order linear equations with variable coefficients,

Euler-Method Cauchy equation;

Determination of complete solution when one solution is known using method of variation of parameters.

Laplace transforms properties and Inverse Laplace properties;

Laplace transforms of elementary functions.

Application to initial value problems for 2nd order linear equations with constant coefficients.

#### (5) Dynamics & Statics:

Rectilinear and Projectile Motion

Motion in a plane, simple harmonic motion, constrained motion;

Work and energy, conservation of energy; Work and potential energy, friction; common catenary;

Central Force Motion: Kepler’s Laws

The principle of virtual work (or principle of virtual displacements)

Equilibrium of a system of particles;

Stability and equilibrium of forces in three dimensions.

#### (6) Vector Analysis:

Vectors and Scalar, fields, differentiation of vector.

Vector Operators: Grad, Div. and Curl

Higher order derivatives; Vector identities and vector equations.

Application to geometry: Curves in space, Serret-Frenet’s formulae, Curvature and torsion;

Gauss and Stokes’ theorems, Green’s identities.

### UPSC Mathematics Optional Syllabus of Paper – II

**(1) Algebra:**

Fundamental algebraic structures, like groups, rings and fields.

Group homomorphism’s, subgroups, cosets, cyclic groups, quotient groups.

Lagrange’s Theorem of group theory, permutation groups, isomorphism theorems and Cayley’s theorem.

Ring homomorphisms and the isomorphism theorems, subrings and ideals.

Principal ideal domain or PID, Integral domains, Euclidean domains and unique factorization domains;

Fields and quotient fields.

**(2) Real Analysis:**

Real number system: Field with The completeness axiom

The limit of a sequence, Cauchy sequence, completeness of real number line;

Convergence of series: Absolute convergence and conditional convergence of all series of real and complex terms, rearrangement of series.

Continuous functions and properties of continuous functions on compact sets.

Riemann integral and improper integrals;

Fundamental theorems of integral calculus.

Uniform convergence and continuity, Integration, differentiation.

Partial derivatives of functions of two or three variables, maxima and minima.

**(3) Complex Analysis:**

Analytic functions, Cauchy’s theorem, Cauchy-Riemann equations, and Cauchy’s integral formula.

Analytic function and Power series representation,

Laurent’s series; Taylor’s series; Singularities

Contour integration, Cauchy’s residue theorem;

**(4) Linear Programming:**

Linear programming, basic problems and solution, optimal solution;

Graphical method and simplex method of solutions; Concept of Duality.

Transportation Problems and assignment problems.

**(5) Partial differential equations:**

Surface and Integral Curves. Three dimensions family, partial differential equations;

Quasilinear partial differential equations of the first order,

Cauchy’s method of characteristics;

Second order differential equations (constant coefficients, canonical form)

Equation of a vibrating string, Equation of heat and Laplace equation.

**(6) Numerical Analysis and Computer Programming:**

Numerical methods: Algebraic functions and transcendental functions,

Methods of Regula-Falsi and Newton-Raphson;

System of linear equations by Gaussian elimination and Gauss-Jordan (direct),

Newton’s (forward and backward) interpolation,

Gauss- Seidel (iterative) methods.

Lagrange’s interpolation.

Numerical integration: Simpson’s rules like 1/3 and 3/8, Trapezoidal rule and errors,

Gaussian quadrature formula.

Ordinary differential equations and numerical solutions

Euler and Runga Kutta-methods.

Computer Programming: Binary number system; Arithmetic and logical operations and their use on numbers; Octal and Hexadecimal systems;

Number System Conversion Rule; Algebra of binary numbers.

Elements of computer systems and memory; Basic logic gates and truth tables, normal forms.

Boolean algebra and its concepts

Signed integers and unsigned integers and

Real’s, double precision real’s and long integers.

Algorithms and flow charts for numerical analysis problems.

**(7) Mechanics and Fluid Dynamics:**

Generalized coordinates; Lagrange’s equations and D’ Alembert’s principle;

Hamilton equations; Motion of rigid bodies in two dimensions, Moment of inertia.

Continuity equation; Euler’s equation; Stream-lines, Potential flow; path of a particle;

Two-dimensional and axi-symmetric motion; vortex motion; sources and sinks, Navier-Stokes equation for a viscous fluid.

UPSC Mathematics Syllabus 2018 is to be covered on a fast pace as very less time is left.