- I.A.S. (MAINS) : MATHEMATICS
- (MORE THAN 80% SYLLABUS COVERED BY RAJNEESH KUMAR SRIVASTAVA)
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- PAPER-I

- Vector spaces over R and C
- Linear dependence and independence
- Subspaces
- Bases, dimension
- Linear transformations
- Rank and nullity
- Matrix of a linear transformation.

- Real numbers
- Functions of a real variable
- Limits, continuity, differentiability
- Mean-value theorem
- Taylor's theorem with remainders
- Indeterminate forms
- Maxima and minima
- Asymptotes
- Curve tracing

- Functions of two or three variables
- Limits, continuity
- Partial derivatives
- Maxima and minima
- Lagrange's method of multipliers
- Jacobian

- Riemann's definition of definite integrals
- Indefinite integrals
- Infinite and improper integrals
- Double and triple integrals (evaluation techniques only)
- Areas, surface and volumes.

- Cartesian and polar coordinates in three dimensions
- Second degree equations in three variables
- Reduction to canonical forms
- Straight lines, shortest distance between two skew lines
- Plane
- Sphere
- Cone
- Cylinder

- Paraboloid
- Ellipsoid
- Hyperboloid of one and two sheets and their properties.

- Formulation of differential equations
- Equations of first order and first degree, integrating factor
- Orthogonal trajectory
- Equations of first order but not of first degree, Clairaut's equation
- Singular solution
- Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution
- Second order linear equations with variable coefficients, Euler-Cauchy equation
- Determination of complete solution when one solution is known using method of variation of parameters.

- Laplace and Inverse Laplace transforms and their properties
- Laplace transforms of elementary functions
- Application to initial value problems for 2nd order linear equations with constant coefficients.

- Rectilinear motion
- Simple harmonic motion
- Motion in a plane
- Projectiles
- Constrained motion, Work and energy
- Conservation of energy
- Kepler's laws, orbits under central forces

- Equilibrium of a system of particles
- Work and potential energy
- Friction
- Common catenary
- Principle of virtual work
- Stability of equilibrium
- Equilibrium of forces in three dimensions.

- Scalar and vector fields
- Differentiation of vector field of a scalar variable
- Gradient, divergence and curl in cartesian and cylindrical coordinates
- Higher order derivatives
- Vector identities and vector equations.
- Gauss and Stokes' theorems, Green's identities.

- Application to geometry
- Curves in space, Curvature and torsion; Serret-Frenet's formulae.

- Groups
- Subgroups
- Cyclic groups
- Cosets, Lagrange's Theorem
- Normal subgroups, quotient groups
- Homomorphism of groups
- Basic isomorphism theorems
- Permutation groups, Cayley's theorem.

- Rings
- Subrings and ideals
- Homomorphisms of rings
- Integral domains
- Principal ideal domains, Euclidean domains and unique factorization domains
- Fields, quotient fields.

- Real number system as an ordered field with least upper bound property
- Sequences
- Limit of a sequence
- Cauchy sequence
- Completeness of real line

- Series and its convergence
- Absolute and conditional convergence of series of real and complex terms
- Rearrangement of series.

- Continuity and uniform continuity of functions
- Properties of continuous functions on compact sets.
- Riemann integral
- Improper integrals
- Fundamental theorems of integral calculus
- Uniform convergence
- Continuity, differentiability and integrability for sequences and series of functions
- Partial derivatives of functions of several (two or three) variables
- Maxima and minima.

- Analytic functions
- Cauchy-Riemann equations
- Cauchy's theorem
- Cauchy's integral formula
- Power series representation of an analytic function
- Taylor's series
- Singularities
- Laurent's series
- Cauchy's residue theorem
- Contour integration.

- Linear programming problems
- Basic solution, basic feasible solution and optimal solution
- Graphical method and simplex method of solutions
- Duality
- Transportation
- Assignment problems.

- Family of surfaces in three dimensions and formulation of partial differential equations
- Solution of quasilinear partial differential equations of the first order
- Cauchy's method of characteristics
- Linear partial differential equations of the second order with constant coefficients
- Canonical form
- Equation of a vibrating string
- Heat equation, Laplace equation and their solutions.

- Numerical methods
- Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods
- Solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods
- Newton's (forward and backward) interpolation, Lagrange's interpolation.