The UPSC Indian Forest Service (IFS) examination offers Mathematics as one of the optional subjects. The syllabus for Mathematics is comprehensive and covers a variety of topics. Here is the detailed syllabus:
Paper-I
1. Linear Algebra:
- Vector spaces, linear dependence and independence, subspaces, bases, dimensions; linear transformations, rank and nullity, matrix representation.
- Systems of linear equations, matrices, eigenvalues, and eigenvectors, Cayley-Hamilton theorem.
2. Calculus:
- Real numbers, limits, continuity, differentiability, mean value theorems, Taylor series.
- Functions of two or three variables, partial derivatives, maxima and minima, Lagrange multipliers.
- Riemann integration, fundamental theorems of calculus, improper integrals.
3. Analytic Geometry:
- Cartesian and polar coordinates, plane, straight lines, sphere, parabola, ellipse, hyperbola.
- Second degree equations, tangent and normal.
4. Ordinary Differential Equations:
- Formulation, order and degree, first-order equations, integrating factors, Bernoulli’s equation.
- Higher-order linear differential equations with constant coefficients, Euler-Cauchy equations.
5. Dynamics and Statics:
- Rectilinear motion, simple harmonic motion, projectiles, circular motion.
- Equilibrium of forces, friction, virtual work, potential energy and stability of equilibrium.
6. Vector Analysis:
- Scalar and vector fields, gradient, divergence, curl, line and surface integrals, Green’s, Gauss’s, and Stokes’s theorems.
7. Algebra:
- Groups, subgroups, normal subgroups, homomorphisms, automorphisms.
- Ring theory, ideals, quotient rings, Euclidean rings, polynomial rings, unique factorization domains.
Paper-II
1. Real Analysis:
- Real number system, sequences, series, convergence, continuity, differentiability.
- Functions of a real variable, Riemann integrals, improper integrals.
2. Complex Analysis:
- Analytic functions, Cauchy-Riemann equations, line integrals, Cauchy’s theorem and integral formula.
- Singularities, residues, contour integration.
3. Linear Programming:
- Linear programming problems, simplex method, duality.
- Transportation and assignment problems.
4. Partial Differential Equations:
- First-order partial differential equations, Cauchy problem for first-order equations.
- Linear second-order partial differential equations, separation of variables, wave, heat and Laplace equations.
5. Numerical Analysis and Computer Programming:
- Numerical solutions of algebraic equations, interpolation, numerical differentiation and integration.
- Numerical solutions of ordinary differential equations, finite differences, Euler and Runge-Kutta methods.
- Computer programming, algorithmic languages.
6. Mechanics and Fluid Dynamics:
- Mechanics of rigid bodies, equations of motion, energy and momentum principles.
- Mechanics of deformable bodies, stress and strain, Hooke’s law.
- Fluid dynamics, equation of continuity, Euler’s equations of motion, Bernoulli’s theorem.
7. Probability and Statistics:
- Probability, random variables, distribution functions, expectation, variance.
- Binomial, Poisson, normal distributions, sampling theory, estimation, hypothesis testing.
This syllabus for Mathematics covers both theoretical and applied aspects, providing a strong foundation for various problems and scenarios that candidates might encounter in the UPSC IFS examination.
