Choosing an optional is a task in itself! Both the papers together account for 500/1750 marks in UPSC Main Examination. Hence, it should be chosen with utmost diligence. As a general rule of thumb, an optional subject should be a subject of your interest. Maths is one of the popular optional subjects for UPSC Main Examination. However, only the aspirants who have studied this subject their graduation in Mathematics should consider opting it.
Some of the benefits of taking up Maths optional include:
UPSC Maths Optional Syllabus  Paper 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’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, CayleyHamilton theorem, Symmetric, skewsymmetric, Hermitian, skewHermitian, 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 integral; 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 liner equations with constant coefficients, complementary function, particular integral and general solution. Section order linear equations with variable coefficients, EulerCauchy 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 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. 
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; SerretFurenet's formulae. Gauss and Stokes’ theorems, Green's identities. 
UPSC Maths Optional Syllabus  Paper 2
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, 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, CauchyRiemann 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, RegulaFalsi and NewtonRaphson methods, solution of system of linear equations by Gaussian Elimination and GaussJorden (direct), GaussSeidel (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: Eular 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  Generalised 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; Streamlines, path of a particle; Potential flow; Twodimensional and axisymmetric motion; Sources and sinks, vortex motion; NavierStokes equation for a viscous fluid. 
Booklist for UPSC Maths Optional
Paper1 
LINEAR ALGEBRA
CALCULUS
ANALYTIC GEOMETRY
ORDINARY DIFFERENTIAL EQUATIONS (ODE)
DYNAMICS AND STATICS
VECTOR ANALYSIS

Paper2 
ALGEBRA
REAL ANALYSIS 
COMPLEX ANALYSIS
LINEAR PROGRAMMING
PARTIAL DIFFERENTIAL EQUATIONS
NUMERICAL ANALYSIS AND COMPUTER PROGRAMMING For Numerical Analysis
For Computer Programming
MECHANICS AND FLUID DYNAMICS

UPSC Maths Optional is a preferred choice of optional for engineering students owing to its objective syllabus. Here’re a few topics that will help you in preparation of UPSC Maths optional:
This was a complete overview of UPSC Maths Optional Syllabus, Booklist and preparation strategy. If you’re still on the fence about your optional selection, you can check out how to choose UPSC Optional subject in the linked article.
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