Analytical Geometry of two and three dimensions:
First and second degree equations in two dimensions in cartesian and polar coordinates. Plane, sphere, paraboloid, Ellipsoid, hyperboloid of one and two sheets and their elementary properties. Curves in space. Curvature and torsion. Frenet's formulze.

Differential Equations :
Order and Degree of a differential equation, differential equation of first order and first degree, variables separable. Homogeneous, linear, and exact differential  quations, differential equations with constant coefficients. The complementary function and the particular integral of eax, cosax, sinax, xm, eax, cosbx, eax, sinbx.

Vector Analysis :
Vector Algebra, Differentiation of vector function of a scalar variable Gradient,divergence and curl in cartesian, cylindrical and spherical coordinates and their physical nterpretation. Higher order derivates. Vector identities and vector, equations, Gauss and stokes Theorems.

Tensor Analysis :
Definition of Tensor, Transformation of coordinates, contravariant and contravariant tensors. Addition and multiplication of tensors, contraction of tensors. Innerproduct, fundamental tensors, Christoffel symbols, contravariant differentiation,Gradiant, curl and divergence in tensor notation.
Statics :
Equilibrium of a system of particles, work and potential energy. Friction. Common catenary. Principle of Virtual work.....Stability of equilibrium. Equilibrium of forces in threedimensions.
Dynamics :
Degree of freedom and constraints. Rectilinear motion Simple harmonic motion in a plane. Projectiles, Constrained motion, work and energy. Motion under impulsive forces. Kepler's laws. Orbits under centralforces. Motion of varying mass. Motion under resisting medium.
Hydrostatics :
Pressure of heavy fluids. Equilibrium of fluids under given system of forces. Centre of pressure. Thrust on curved surfaces. Equilibrium of floating bodies, stability of equilibrium and pressure and gases, problems relating to atmosphere.Paper - II Algebra:
Groups, subgroups, normal subgroups, homomorphism of groups, quotient groups Basic isomorphism theorems, sylow theorems. Permutation Groups. Cayley's Theorem. Rings and ideals, Principal ideal domains, unique factorization domains and Euciidean domains, Field Extensions, Finite fields.
Real Analysis:
Metric spaces, their topology with special reference to ‘R' sequence in metric space Cauchy sequence completeness. Completion, continuous functions. Uniform continuity. Properties of continuous function on Compact sets. Riemann Steltjes Integral. Improper integrals and their condition's of existence. Differentiation of function of several variables. Implicit function theorem, maxima and minima. Absolute and conditional Convergence of series of real Complex terms, Rearrangement of series. Uniform -convergence, infinite products. Continuity, differentiability and integrability for series, Multiple integrals.
Complex Analysis:
Analytic functions, Cauchy's theorem, Cauchy's integral formula, power series, Taylor's series. Singularities, Cachley's Residue theorem and Contour integration.
Partial Differential Equations:
Formation of partial differential equations. Types of integrals of partial differential equations of first order, Charphs method, Partial differential equation with constant coefficients.
Mechanics:
Generalised Coordinates, constraints, holonomic and non-holonomic systems, D ‘Alemberts' Principle and Langrange's equations, Moment of inertia. Motion of rigid bodies in two dimensions.
Hydrodynamics:
Equation of continuity, momentum and energy, inviscid flow theory, Two dimensional motion, streaming motion sources and Sinks.
Numerical Analysis:
Transcendental and polynomial Equations-Methods of tabulation, bisection, reaula-false secants and Newton-Renhsonand order of its converagence. Interpolation and Numerical differentiation Polynomial interpolation with equal or unequal step size. Numerical differentiation formulae with error terms.
Numerical integration of Ordinary differential Equations:
Euler's method, mulistepperdictors Corrector methods. Adam's and Milne's method convergence and stability, Runge Kutta Methods. Operational Research:
Mathematical Programming, Definition and some elementary properties of convex sets, simplex methods, rectangular games and their solutions.