The Mathematics course is handled by Mr. Venkanna . One of the renouned name for MATHEMATICS optional for UPSC in Delhi. Venkanna Sir is very well known for producing top rankers of UPSC with MATHEMATICS optional,Delhi

Mathematics Optional Syllabus

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

MATHEMATICS SYLLABUS for PAPER - 1

1. Linear Algebra

   Venkanna Sir

   40 Hrs

   • 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.

2. Calculus

   Venkanna Sir

   40 Hrs

   • 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.

3. Analytic Geometry

   Venkanna Sir

   30 Hrs

   • 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.

4. Ordinary Differential Equations

   Venkanna Sir

   20 Hrs

   • 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.

5. Dynamics & Statics

   Venkanna Sir

   35 Hrs

   • 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.

6. Vector Analysis

   Venkanna Sir

   25 Hrs

   • 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 for PAPER 2

1. Abstract Algebra

   Venkanna Sir

   40 Hrs

   • 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.

2. Real Analysis

   Venkanna Sir

   40 Hrs

   • 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.

3. Complex Analysis

   Venkanna Sir

   15 Hrs

   • 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.

4. Linear Programming

   Venkanna Sir

   15 Hrs

   • Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality.
   • Transportation and assignment problems.

5. Partial Differential Equations

   Venkanna Sir

   20 Hrs

   • 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.

6. Numerical Analysis and Computer Programming

   Venkanna Sir

   15 Hrs

   • 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 rules, Gaussian quadrature formula.
   • 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.

7. Mechanics and Fluid Dynamics

   Venkanna Sir

   30 Hrs

   • 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 350 hours.
• Click here to check the weekly timetable schedule for the course.
• Click here for Sample Videos.

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.

• Abstract Algebra : 1). Abstarct Algebra by V K Khanna,S K Bhambri 2). Abstract Algebra by P.B Bhattacharya
• Linear Algebra : 1). Bsc Mathematics Volume - III by S Chand 2). Matrices and Linear Algebra by S Chand
• Real Analysis : 1). Elements of Real Analysis by Raisinghania of S Chand
• Solid Geometry : 1). Analytical Geomtry by S Chand, Authors : Shanthi Narayan, P.K Mittal
• Linear Programming : 1). Operations Reasearch by Panner Selvam
• Numerical Methods : 1). Numerical Methods with Computer orientation by any Author 2). Finet Differrences and Numerical Analysis by Gupta of S Chand Or T K V Iyengar
• Ordinary and Partial Differrential Equations : 1). Ordinary and Partial Differrential Equations written by M D Raisinghania
• ODE,PDE, Laplace Transforms,Vector Calculus,Complex Analysis,Calculus : 1). Higher Engineering Mathematics by B S Grewal
• Statics : 1). Krishna Series, A R Vasista and R K Gupta
• Dynamics : 1). Krishna Series, A R Vasista and R K Gupta
• Fluid Dynamics: 1). Shanthis Swaroop - Krishna Prkashan Publishers 2). Or, Fluid Dynamics by M D Raisinghania
• Mechanics : 1). Material Prepared by Venkanna Sir

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 ALL SUBJECTS 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.

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