Syllabus for Paper I of UPSC Mathematics Optional subject:

The syllabus for Paper I of UPSC Mathematics Optional subject is divided into several topics. Here is a detailed breakdown of the syllabus:

1. Linear Algebra:
- Vector spaces, subspaces, linear dependence, basis, dimension, rank and nullity, linear transformations, range and null space, matrix representation, change of basis, eigenvalues and eigenvectors, diagonalization, inner product spaces, orthogonality, Gram-Schmidt process, orthogonal projections and their properties.

2. Calculus:
- Real numbers, limits, continuity, differentiability, mean value theorem, Taylor's theorem, maxima and minima, indeterminate forms, evaluation of definite integrals, improper integrals, double and triple integrals, sequences and series, power series, Fourier series, line, surface and volume integrals, Green's theorem, Stokes' theorem, divergence theorem.

3. Analytic Geometry:
- Cartesian and polar coordinates in three dimensions, distance formula, equations of lines and planes, sphere, cone, cylinder, paraboloid, hyperboloid of one and two sheets, ellipsoid, equations of tangent and normal, orthographic and perspective projections.

4. Ordinary Differential Equations (ODEs):
- First-order ODEs, linear and nonlinear ODEs, homogeneous and non-homogeneous ODEs, exact and reducible to exact form, integrating factors, Bernoulli's equation, linear equations of higher order with constant coefficients, Cauchy-Euler equation, power series solutions, Legendre's equation, Bessel's equation, Laplace transforms, inverse transforms, applications to initial and boundary value problems.

5. Dynamics and Statics:
- Rectilinear motion, simple harmonic motion, forced and damped oscillations, projectile motion, central forces, Kepler's laws, motion under inverse square law, work and energy, conservation of energy, conservative forces, potential energy, equilibrium of particles and rigid bodies, friction, virtual work.

6. Vector Analysis:
- Scalar and vector fields, derivatives and integrals of scalar and vector fields, line, surface and volume integrals, Stokes' theorem, Green's theorem, Gauss' divergence theorem.

7. Algebra:
- Groups, subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups, quotient groups, homomorphism and isomorphism theorems, permutation groups, group actions, Sylow's theorems, rings, subrings, integral domains, fields, polynomial rings, Euclidean domains, principal ideal domains, unique factorization domains, fields of fractions, quotient fields.

8. Real Analysis:
- Real number system, completeness axiom, sequences and series of real numbers, limits, continuity, differentiability, mean value theorem, Riemann integration, improper integrals, sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Green's theorem, Stokes' theorem, divergence theorem.

9. Complex Analysis:
- Complex numbers, complex plane, modulus and argument, polar form, triangle inequality, logarithm and exponential functions, analytic