**Mathematics Optional Course For UPSC**

Choosing the right object for the preparation of the UPSC examination is a crucial decision that a candidate has to make. The main rule for selecting an optional subject is that a candidate should take the subject in which you have some academic background and real interest. along with this, there are some other factors that should be considered while choosing a subject as an optional subject for the preparation of UPSC civil service examination that includes the past performance of the subject, the time frame required to prepare for the subject, the contribution of the subject towards the general studies paper and the coaching institute available for the preparation of the optional subject that you have selected. All these things also apply for opting for mathematics as an optional subject for the preparation of the UPSC examination. Firstly, a candidate should know that math is a subject that can be scored only if you have a proper understanding of the formulas and the concepts. The candidate who is using mathematics must have studied the subject in their graduation as math till 12th is not enough for the preparation of the UPSC civil service examination. The candidate must have a good interest in mathematics to score good marks in the UPSC civil service examination.

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**What is the syllabus of mathematics optional subject for UPSC in the UPSC mains examination?**

The optional subject has two papers: paper I and paper II. Every paper of the optional subject is of 250 marks, which makes it a total of 500 mark paper.

Following is the syllabus for both the papers of math optional as per the UPSC official notification:-

**Mathematics paper I**

**Linear algebra**

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, 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 the system of linear equations, Eigenvalues and eigenvectors, characteristic polynomial, orthogonal and unitary matrices, and their eigenvalues.

**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, Jacobian. Riemann’s definition of definite integrals, Indefinite integrals, Infinite and improper integral, Double and triple integrals (evaluation techniques only), Curve tracing, Functions of two or three variables, Limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Areas, surface, and volumes.

**Analytical geometry**

Cartesian and polar coordinates in three dimensions, second-degree equations in three variables, reduction to Canonical forms, straight lines, the 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 the first order and first degree, integrating factor, Orthogonal trajectory, Equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, Laplace and Inverse Laplace transform and their properties, Laplace transforms of elementary functions. Application to initial value problems for second-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 the method of variation of parameters.

**Dynamics and static**

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 equation. Application to geometry: Curves in space, curvature, and torsion, Serret-Furenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

**Mathematics paper II**

**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.

**Real analysis **

Real number system as an ordered field with least upper bound property, Sequences, the 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 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.

**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 a 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 Runge 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, 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, the path of a particle, Potential flow, Two-dimensional and axisymmetric motion, Sources and sinks, vortex motion, Navier-Stokes equation for a viscous fluid.

**What are the benefits of taking math as an optional subject for the UPSC main examination?**

**Following are the benefits of choosing mathematics subject of UPSC examination:- **

- Choosing mathematics as an optional subject can be very helpful for the students as the subject has a static syllabus.
- The well-balanced syllabus helps the candidates in understanding well.
- This subject has high objectivity because of which it is a scoring subject.
- This paper has less emphasis on memory.
- This paper requires comparatively less time to prepare for this subject.

**What are the tips to score more marks in the math optional subject?**

Our teachers were absolutely right about the fact that if you want to score more in a math examination you have to practice more and avoid silly mistakes. there are some of the basic keys to scoring high in the mathematics subject that includes more practice here are some of the tips that can be helpful for the candidates to help them in scoring in the math optional subject:-

- The candidates should practice more to score good marks in the examination.
- the candidate should avoid silly mistakes.
- Math is a logical subject so the candidate should never try to cram the theorems and solutions of the subject.
- The candidates should be systematic and the presentation of the solution should be smooth.
- The candidate should always keep a formula sheet with them so that they can revise anywhere they want to.