Mathematics Syllabus for competitive exams and gate

Section  1:  Linear Algebra

  Finite dimensional  vector  spaces;  Linear  transformations  and  their  matrix representations, rank;  systems  of  linear  equations,  eigenvalues  and  eigenvectors,  minimal  polynomial, Cayley-Hamilton  Theorem,  diagonalization,  Jordan-canonical  form,  Hermitian,  SkewHermitian  and  unitary  matrices;  Finite dimensional  inner  product  spaces,  Gram-Schmidt orthonormalization  process,  self-adjoint  operators,  definite forms.

Section 2: Complex Analysis 

Analytic  functions,  conformal  mappings,  bilinear  transformations;  complex  integration: Cauchy’s  integral  theorem  and  formula;  Liouville’s  theorem,  maximum  modulus  principle; Zeros  and  singularities;  Taylor  and  Laurent’s  series;  residue theorem  and  applications  for evaluating  real  integrals.

Section 3: 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, compactness,  completeness,  Weierstrass  approximation  theorem;  Lebesgue measure, measurable functions;  Lebesgue integral,  Fatou’s  lemma,  dominated  convergence theorem.

Section  4: Ordinary  Differential  Equations 

First  order  ordinary  differential  equations,  existence and  uniqueness  theorems  for  initial value  problems,  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  (power  series,  Frobenius  method); Legendre and  Bessel functions  and  their  orthogonal  properties.

Section  5:  Algebra

Groups,  subgroups,  normal  subgroups,  quotient  groups  and  homomorphism  theorems, automorphisms;   cyclic  groups  and  permutation  groups,  Sylow’s  theorems  and  their applications;   Rings,  ideals,  prime and  maximal  ideals,  quotient  rings,  unique factorization domains,  Principle ideal  domains,  Euclidean domains,  polynomial  rings  and  irreducibility criteria;  Fields,  finite fields,  field  extensions.

Section  6: Functional  Analysis

  Normed  linear  spaces,  Banach  spaces,  Hahn-Banach  extension theorem,  open mapping and  closed  graph  theorems,  principle  of  uniform  boundedness;  Inner-product  spaces, Hilbert  spaces,  orthonormal  bases,  Riesz  representation  theorem,  bounded  linear operators.

Section  7:  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;  numerical  solution  of  systems  of  linear equations:  direct  methods  (Gauss  elimination,  LU  decomposition);  iterative  methods :Jacobi  and  Gauss-Seidel);  numerical  solution of  ordinary  differential  equations:  initial value problems:  Euler’s  method,  Runge-Kutta  methods  of  order  2.

Section  8:  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 in  two  dimensional  Cartesian coordinates,  Interior  and  exterior  Dirichlet  problems  in polar  coordinates;  Separation of variables  method  for  solving  wave and  diffusion equations  in one space  variable;  Fourier series  and  Fourier  transform  and  Laplace  transform  methods  of  solutions  for  the  above equations.

Section  9: Topology

Basic  concepts of  topology,  bases,  subbases,  subspace  topology,  order  topology, product  topology,  connectedness,  compactness,  countability  and  separation axioms, Urysohn’s Lemma.

Section  10: Probability  and  Statistics Probability  space,  conditional  probability,  Bayes  theorem,  independence,  Random variables,  joint  and  conditional  distributions,  standard  probability  distributions  and  their properties  (Discrete  uniform,  Binomial,  Poisson,  Geometric,  Negative binomial,  Normal, Exponential,  Gamma,  Continuous  uniform,  Bivariate normal,  Multinomial),  expectation, conditional  expectation,  moments;  Weak  and  strong  law  of  large numbers,  central  limit theorem;  Sampling  distributions,  UMVU  estimators,  maximum  likelihood  estimators;  Interval estimation;  Testing  of  hypotheses,  standard  parametric  tests  based  on  normal, , ,    distributions;  Simple  linear  regression.

 Section  11:  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,  Vogel’s  approximation method  for  solving transportation problems;  Hungarian method  for  solving  assignment  problems. 

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