Monday, 3 October 2016

RGPV TET Mathematics Syllabus | Download RGPV TET Engg mATHS Syllabus www.rgpv.ac.in

RGPV TET Mathematics Syllabus | Download RGPV TET Engg mATHS Syllabus

RGPV TET Mathematics Syllabus | Download RGPV TET Engg mATHS Syllabus


Linear  Algebra: Finite  dimensional  vector  spaces;  Linear  transformations  and  their  matrix
representations,  rank;  systems  of  linear  equations,  eigen  values  and  eigen  vectors,  minimal
polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian, Skew-Hermitian and unitary
matrices;  Finite  dimensional  inner  product  spaces,  Gram-Schmidt  orthonormalization  process,
self-adjoint operators.


Complex  Analysis: Analytic functions, conformal mappings, bilinear transformations; complex
integration:  Cauchy’s  integral  theorem  and  formula;  Liouville’s  theorem,  maximum  modulus
principle;  Taylor  and  Laurent’s  series;  residue  theorem  and  applications  for  evaluating  real
integrals.
Real  Analysis: Sequences and series of functions, uniform convergence, power series, Fourier
series, functions of several variables, maxima, minima; Riemann integration, multiple integrals,
line,  surface  and  volume  integrals,  theorems  of  Green,  Stokes  and  Gauss;  metric  spaces,
completeness, Weierstrass  approximation theorem, compactness; Lebesgue measure, measurable
functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem.
Ordinary  Differential  Equations: First  order  ordinary  differential  equations,  existence  and
uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary
differential  equations  of  higher  order  with  constant  coefficients;  linear  second  order  ordinary
differential  equations  with  variable  coefficients;  method  of  Laplace  transforms  for  solving
ordinary  differential  equations,  series  solutions;  Legendre  and  Bessel  functions  and  their
orthogonality.

/div>
Algebra:Normal  subgroups  and  homomorphism  theorems,  automorphisms;  Group  actions,
Sylow’s theorems and their applications; Euclidean domains, Principle ideal domains and unique
factorization  domains.  Prime  ideals  and  maximal  ideals  in  commutative  rings;  Fields,  finite
fields.
Functional  Analysis:Banach spaces, Hahn-Banach extension theorem, open mapping and closed
graph  theorems,  principle  of  uniform  boundedness;  Hilbert  spaces,  orthonormal  bases,  Riesz
representation theorem, bounded linear operators.
Numerical  Analysis: Numerical  solution  of  algebraic  and  transcendental  equations:  bisection,
secant method, Newton-Raphson method, fixed point iteration; interpolation: error of polynomial
interpolation, Lagrange, Newton interpolations; numerical differentiation; numerical integration:
Trapezoidal and Simpson rules, Gauss Legendrequadrature, method of undetermined parameters;
least square polynomial approximation; numerical solution of systems of linear equations: direct
methods (Gauss elimination,  LU decomposition); iterative methods (Jacobi and Gauss-Seidel);
matrix  eigenvalue  problems:  power  method,  numerical  solution  of  ordinary  differential
equations: initial value problems: Taylor series methods, Euler’s method, Runge-Kutta methods.
Partial  Differential  Equations: Linear and quasilinear first order partial differential equations,
method of characteristics; second order linear equations in two variables and their classification;
Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in
two variables; Fourier series and Fourier transform and Laplace transform methods of solutions
for the above equations.
Mechanics: Virtual work, Lagrange’s equations for holonomic systems, Hamiltonian equations.
Topology: Basic  concepts  of  topology,  product  topology,  connectedness,  compactness,
countability and separation axioms, Urysohn’s Lemma.
Probability  and  Statistics: Probability  space,  conditional  probability,  Bayes  theorem,
independence,  Random  variables,  joint  and  conditional  distributions,  standard  probability
distributions  and  their  properties,  expectation,  conditional  expectation,  moments;  Weak  and
strong law of large  numbers, central limit theorem; Sampling distributions, UMVU estimators,
maximum  likelihood  estimators,  Testing  of  hypotheses,  standard  parametric  tests  based  on
normal, X
2
, t, F – distributions; Linear regression; Interval estimation.

Linear  programming: Linear programming problem and its formulation, convex sets and their
properties,  graphical  method,  basic  feasible  solution,  simplex  method,  big-M  and  two  phase
methods; infeasible and unbounded LPP’s, alternate optima; Dual problem and duality theorems,
dual  simplex  method  and  its  application in  post optimality  analysis;  Balanced  and  unbalanced
transportation problems, u  -u method for solving transportation problems; Hungarian method for
solving assignment problems.
Calculus  of  Variation  and  Integral  Equations: Variation  problems  with  fixed  boundaries;
sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their
iterative solutions.

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