# (Paper) Mathematics : Main Examination

**UPSC
: Mathematics : Main Examination
**

**PAPER-I**

**Section-A**

**Linear Algebra** :
Vector, space, linear dependance and independance, subspaces, bases, dimensions.
Finite dimensional vector spaces.

Matrices, Cayley-Hamiliton theorem, eigenvalues and eigenvectors, matrix of linear transformation, row and column reduction, Echelon form, eqivalence, congruences and similarity, reduction to cannonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian forms-their eigenvalues. Orthogonal and unitary reduction of quadratic and hermitian forms, positive definite quardratic forms.

**Calculus**: Real
numbers, limits, continuity, differerentiability, mean-value theorems, Taylors
theorem with remainders, indeterminate forms, maximas and minima, asyptotes.
Functions of several variables: continuity, differentiability, partial
derivatives, maxima and minima, Lagranges method of multipliers, Jacobian.
Riemanns definition of definite integrals, indefinite integrals, infinite and
improper intergrals, beta and gamma functions. Double and triple integrals
(evaluation techniques only). Areas, surface and volumes, centre of gravity.

**Analytic Geometry:**Cartesian
and polar coordinates in two and three dimesnipns, second degree equations in
two and three dimensions, reduction to cannonical forms, straight lines,
shortest distance between two skew lines, plane, sphere, cone, cylinder.,
paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

**Section-B**

**Ordinary Differential
Equations:**Formulation of differential equations, order and degree,
equations of first order and first degree, integrating factor, equations of
first order but ndt of first degree, Clariauts equation, singular solution.

Higher order linear equations, with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation.

Second order linea: equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.

**Dynamics, Statics and
Hydrostatics:**Degree of freedom and constraints, rectilinerar motion,
simple harmonic motion, motion in a plane, projectiles, constrained motion, work
and energy, conservation of energy, motion under impulsive forces, Keplers laws,
orbits under central forces, motion of varying mass, motion under resistance.
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.

Pressure of heavy fluids, equilibrium of fluids under given system of forces Bernoullis equation, centre of pressure, thrust on curved surfaces, equilibrium of floating bodies, stability of equilibrium, metacentre, pressure of gases.

**Vector Analysis :**Scalar
and vector fields, triple, products, differentiation of vector function of a
scalar variable, Gradient, divergence and curl in cartesian, cylindrical and
spherical coordinates and their physical interpretations. Higher order
derivatives, vector identities and vector quations.

Application to Geometry: Curves in space, curvature and torision. Serret-Frenets formulae, Gauss and Stokes theorems, Greens identities.

**PAPER-II**

**Section-A**

**Algebra:**Groups,
subgroups, normal subgroups, homomorphism of groups quotient groups basic
isomorophism theorems, Sylows group, permutation groups, Cayley theorem. Rings
and ideals, principal ideal domains, unique factorization domains and Euclidean
domains. Field extensions, finite fields.

**Real Analysis :**Real
number system, ordered sets, bounds, ordered field, real number system as an
ordered field with least upper bound property, cauchy sequence, completeness,
Continuity and uniform continuity of functions, properties of continuous
functions on compact sets. Riemann integral, improper integrals, absolute and
conditional convergence of series of real and complex terms, rearrangement of
series. Uniform convergence, continuity, differentiability and integrability for
sequences and series of functions. Differentiation of ructions of several
variables, change in the order of partial derivatives, implict function theorem,
maxima and minima. Multiple integrals.

**Complex Analysis :**
Analytic function, Cauchy-Riemann equations, Cauchys theorem, Cauchys integral
formula, power series, Taylors series, Laurents Series, Singularities, Cauchys
residue theorem, contour integration. Conformal mapping, bilinear
transformations.

**Linear Programming :**Linear
programming problems, basic solution, basic feasible solution and optimal solu
tion, graphical method and Simplex method of solutions. Duality. Transportation
and assignment problems. Travelling salesman problmes.

**Section-B**

**Partial differential
equations:** Curves and surfaces in three dimesnions, formulation of
partial differential equations, solutions of equations of type dx/p=dy/q=dz/r;
orthogonal trajectories, pfaffian differential equations; partial differential
equations of the first order, solution by Cauchys method of characteristics;
Charpits method of solutions, linear partial differential equations of the
second order with constant coefficients, equations of vibrating string, heat
equation, laplace equation.

**Numerical Analysis and
Computer programming:** 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) methods, Gauss-Seidel(iterative) method.

Newtons (Forward and backward) and Lagranges method of interpolation.

Numerical integration: Simpsons one-third rule, tranpezodiai rule, Gaussian quardrature formula.

Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.

**Computer Programming:**
Storage of numbers in Computers, bits, bytes and words, binary system,
arithmetic and logical operations on numbers. Bitwise operations. AND, OR, XOR,
NOT, and shift/rotate operators. Octal and Hexadecimal Systems. Conversion to
and form decimal Systems.

Representation of unsigned integers, signed integers and reals, double precision reals and long integrers.

Algorithms and flow charts for solving numerical analysis problems.

Developing simple programs in Basic for problems involving techniques covered in the numerical analysis.

**Mechanics and Fluid
Dynamics :** Generalised coordinates, constraints, holonomic and
non-holonomic, systems. D Alemberts principle and Lagrange equations, Hamilton
equations, moment of intertia, motion of rigid bodies in two dimensions.

Equation of continuity, Eulers equation of motion for inviscid flow, streamlines, path of a particle, potential flow, two-dimensional and axisymetric motion, sources and sinks, vortex motion, flow past a cylinder and a sphere, method of images. Navier-Stokes equation for a viscous fluid.