The syllabus for the Mathematics optional paper in the UPSC Civil Services Examination is relatively extensive. Here is a broad overview of the syllabus:
Paper-I:
- Linear Algebra:
- Vector spaces over R and C, matrices, determinants, and their properties, eigenvalues, eigenvectors, Cayley-Hamilton theorem.
- Calculus:
- Real Analysis, sequences and series of functions, continuity, differentiability, mean value theorems, and applications, convergence and divergence of improper integrals.
- Analytic Geometry:
- Cartesian and polar coordinates, the equation of a line, plane, conic section in Cartesian coordinates, and polar coordinates.
- Ordinary Differential Equations (ODE):
- First-order equations (linear and nonlinear), higher-order linear differential equations, Cauchy’s, and Euler’s equations, methods of solution using variations of parameters, and Laplace transforms.
- Dynamics & Statics:
- Rectilinear motion, simple harmonic motion, Kepler’s laws, the central force in a plane, collisions, and moment of inertia, the system of particles, and the concept of rigid body.
Paper-II:
- Algebra:
- Groups, subgroups, normal subgroups, permutation groups, cyclic groups, Lagrange’s theorem, Sylow theorems, and their applications.
- Real Analysis:
- Lebesgue measure and integration, Banach spaces, Hilbert spaces, Lp spaces, and their properties.
- Complex Analysis:
- Analytic functions, Cauchy-Riemann equations, contour integration, the residue theorem, and applications.
- Partial Differential Equations (PDE):
- Linear and quasi-linear first-order PDE, Cauchy’s method of characteristics, second-order linear PDE, classification, and canonical forms.
- Numerical Analysis and Computer Programming:
- Numerical methods for linear and nonlinear algebraic equations, numerical integration, and differentiation, interpolation, and extrapolation, solutions of ordinary differential equations, finite difference methods.
It is advisable to go through the detailed syllabus provided by the UPSC and refer to standard textbooks for each topic. Additionally, candidates should practice solving previous years’ question papers and participate in mock tests to enhance their preparation.