The following is the syllabus for Mathematics - Main Examination - Paper I and
Paper II.

MATHEMATICS SYLLABUS for PAPER - 1

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,
meanvalue 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.

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 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.

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.

Online Classroom Sessions

The course comprises of online classroom sessions for a duration of 225 hours.

Click here to check the weekly timetable schedule for the course.

Mathematics Optional Suggested Books for Reference

The following are the suggested books / references that the students can consult
while preparing for Mathematics - Main Examination - Paper I and Paper II.

Higher Engineering Mathematics,by B. S. Grewal, Publisher: Khanna Publishers
(2006)

Real Analysis - A. R. Vasishtha and Hemalatha Vasishtha, Publisher: Krishna Prakashan
Media Pvt. Ltd.

Statics - M. L. Khanna, Publisher: Jayaprakash Nath & Co.

Linear Algebra - Kenneth Hoffman, Ray Kunze, Publisher: Prentice Hall of India Pvt.
Ltd. New Delhi

Coordinate Geometry - H. K. Dass, C. H. Saxena, M. D. Raisinghania, Publisher: S.
Chand & Co. Ltd.

Popularity of Mathematics as an optional subject

The following are the suggested books / references that the students can consult
while preparing for Mathematics - Main Examination - Paper I and Paper II.

Mathematics is best fit as an optional for students who are passionate about the
subject.

Since this is logical subject, scoring is quite straightforward and rote-learning
is not required.

Engineering Mathematics Students can score very well in this subject as only standard
models of sums are asked in the question papers.

Suggestions for Preparation of Mathematics Optional Subject

The following are simple preparation tips to score well in the Mathematics -
Main Examination - Paper I and Paper II.

Prepare selective topics and be thorough in these problems.

Linear Algebra, Linear Programming, Vector Calculus, Numerical Methods, Real and
Complex Analysis, Ordinary Differential Equations and Partial Differential Equations
are some examples of topics thay are relatively easy for most people.

Solve problems from previous years questions papers as the same models of the sums
normally repeat every year.

Solve each problem highlighting each step while arriving at the solution. Do not
skip any step thinking it is trivial.

Concentrated on the solved examples from B S Grewal Reference book suggested for
the UPSC exam.

Monitor your speed during practice and try to improve it. Lack of time is the main
challenge in this paper.

Joining the course

Regular Student

Students can register as a regular user for any of the courses offered by Indiancivils
by making the full payment for the course online.

As a regular user, students can

Gain access to the question bank of ALLSUBJECTS
of CSAT (Prelims) .

Gain access to the of the registered courses
for 18 Months.

Registered students would receive intimation about the
scheduled virtual classroom sessions on their registered mobile and
email ids. Further, an automated reminder would be
sent to the registered mobile number 10 minutes
before the online session.