Syllabus for Paper II of UPSC Mathematics Optional subject:

The Paper II of UPSC Mathematics Optional subject consists of the following topics:

1. Linear Algebra:
- Vector spaces, subspaces, linear dependence, basis, dimension, linear transformations, rank and nullity, matrix of a linear transformation.
- Algebra of matrices, rank, inverse, systems of linear equations.
- Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, diagonalization, symmetric, skew-symmetric, Hermitian, skew-Hermitian forms, positive definite, negative definite and indefinite matrices.

2. 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 derivatives, Euler's theorem, tangent plane, maxima and minima, Lagrange's method of multipliers, Jacobian.

3. Analytic Geometry:
- Cartesian and polar coordinates, equations of lines, slopes, angles between lines, equations of circles, parabola, ellipse and hyperbola in standard form.
- General equation of second degree, plane, sphere, cone, cylinder, paraboloid, ellipsoid and hyperboloid in standard form.

4. Ordinary Differential Equations:
- Formulation of differential equations, order and degree, solutions of differential equations.
- Formation of differential equations, solutions of differential equations by various methods, linear differential equations of higher order with constant coefficients.

5. Dynamics and Statics:
- Rectilinear motion, simple harmonic motion, motion in a plane, projectiles.
- Work and energy, conservation of energy, potential energy, conservative forces.
- Equilibrium of a system of particles, friction, laws of friction, equilibrium of a system of particles, center of mass, moment of inertia, radius of gyration.

Summary:
The Paper II of UPSC Mathematics Optional subject covers topics such as linear algebra, calculus, analytic geometry, ordinary differential equations, dynamics, and statics. Students are expected to have a deep understanding of these concepts and their applications. It is important to study and practice each topic thoroughly to perform well in the examination.