The syllabus for the Mathematics Optional in the UPSC exam covers various topics that are important in the field of mathematics. Here is a detailed breakdown of the topics covered in the syllabus:

Linear Algebra:
- Vector spaces, subspaces, linear dependence, basis, dimension, linear transformations, rank and nullity, matrix of a linear transformation.
- Algebra of matrices, row and column reduction, echelon form, congruence, and similarity.
- Determinants, inverse, rank and system of linear equations.

Calculus:
- Real numbers, limits, continuity, differentiability, mean value theorem, Taylor’s theorem, maxima and minima, indeterminate forms, evaluation of definite and improper integrals.
- Partial derivatives, total derivative, maxima and minima of functions of several variables, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes, and Gauss.

Analytic Geometry:
- Cartesian and polar coordinates in three dimensions, distance between two points, the section formula, the direction cosines and direction ratios of a line joining two points, the equation of a plane, a distance of a point from a plane.
- The angle between two lines, the equation of a sphere, a tangent plane, a normal to a sphere, the intersection of a sphere with a plane.

Ordinary Differential Equations (ODE):
- First order ODEs (linear and nonlinear), higher-order linear ODEs with constant coefficients, the method of variation of parameters, the Cauchy-Euler equation.
- Second-order linear homogeneous ODEs with constant coefficients.

Partial Differential Equations (PDE):
- Classification of second-order PDEs, the method of separation of variables for solving homogeneous PDEs, the wave equation, the heat equation, Laplace’s equation.

Vector Analysis:
- Scalar and vector fields, the gradient, divergence, curl, line, surface and volume integrals, Stokes’ theorem, Green’s theorem, the Gauss divergence theorem.

Numerical Analysis and Computer Programming:
- Numerical methods: solution of algebraic and transcendental equations of one variable, interpolation, numerical differentiation and integration, numerical solutions of ODEs.
- Computer programming in C: basics, control statements, functions, arrays, pointers, and structures.

These topics cover a wide range of mathematical concepts and techniques, providing a strong foundation in the subject. It is important for aspirants to thoroughly study and understand these topics in order to excel in the Mathematics Optional paper of the UPSC exam.