**How To Prepare UPSC Mathematics Optional Subject 2022 for Mains Exam**

**How To Prepare UPSC Mathematics Optional Subject 2022 for Mains Exam**

Choosing an optional subject can be an overwhelming exercise. After all, the optional subject contributes 500 marks to your final result making it an essential factor in deciding your success in the examination.

Mathematics as an optional subject is preferred by students because it is highly logical in nature. Students who have a verified interest in mathematics would find it to be a nice break from other GS subjects. If you have completed degree-level examinations with mathematics as a subject and enjoy learning this domain, then choosing mathematics as your UPSC optional may be a wise decision.

Before making the decision to take mathematics as an optional subject it is wise to go through the syllabus and past year papers. The syllabus of mathematics has been shared for your reference.

**Mathematics Optional Syllabus**

**(1) Linear Algebra :**

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.

Algebra of Matrices; Row and column reduction, Echelon form, congruence 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 :**

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 integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

**(3) 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.

**(4) 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.

Section 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 and 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.

**(6) 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 equation.

**Application to geometry : **

Curves in space, curvature and torsion; Serret-Furenet's formulae. Gauss and Stokes’ theorems, Green's identities.

**PAPER II**

**(1) 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.

**(2) 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.

**(3) Complex Analysis :**

Analytic function, 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 :**

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 :**

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 : **

**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) and interpolation, Lagrange’s interpolation.

Numerical integration: Trapezoidal rule, Simpson’s rule, 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 :**

**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.

**The most effective method to Prepare for Mathematics Optional Subject For UPSC**

**What are the upsides of taking math as an optional subject? **

**Scoring Subject **

Mathematics, unlike theoretical subjects, is numeral based that requires clarity of concepts only. If you are good at maths then there is no need to worry much as you can solve the questions easily.

**Straightforward **

The proposed questions in this paper are usually directly derived from the reading material. Any mathematics graduate should be able to answer questions of a low to the medium difficulty level easily.

**Static Syllabus **

This subject's syllabus is set in stone. It has nothing to do with contemporary events. As a result, once you've finished the syllabus, you won't need to refresh your information on a regular basis; all you'll need to do is revise.

**Mathematics Optional Preparation Tips**

**Stage 1: Know UPSC Mathematics Complete Syllabus **

The first step for any exam is going through the syllabus as it is the syllabus that tells you what to read and what not. So, you need to know the whole syllabus first to ensure good planning and preparation for the exam.

**Stage 2: Select Best Reference Books for UPSC Mathematics **

- Calculus and Real Analysis by S.C Malik and Savita Arora, Shanti Narayana
- Linear Algebra by A.R.Vasista
- Numerical Analysis by K. Shankar Rao
- Ordinary Differential Equations by M.D. Raisinghania
- 3-D Geometry by P.N. Chatterjee
- Algebra by Shramik Sen Upadhayay
- Linear Programming by Shanti Swarup, S D Sharma

**Stage 3: Prepare a Study Plan **

Create an examination strategy for the UPSC exam. Make a month-by-month schedule, a week-by-week plan, and a daily schedule to prepare for the UPSC Mathematics Optional Paper.

**Step 4: Cover all important topics**

Some important topics are:

- Statics & Dynamics
- Calculus
- Analytical Geometry
- Abstract Algebra
- Linear Programming
- Mechanics and Fluid Dynamics, etc

**Step 5: Solve Previous Year Paper**

The final stage in completing your UPSC Mathematics Optional Subject 2022 preparation is to routinely practice a large number of prior year problems. Solving last ten years papers is the most effective way to earn good marks.