Integrated Ph.D Entrance Exam Maths Syllabus

I am looking for Integrated Ph.D Entrance Exam Mathematics Syllabus, will you please provide here???

[Kiran Chandar]

You are looking for Integrated Ph.D Entrance Exam Mathematics Syllabus, here I am providing:

Set Theory: Sets, operations on sets, De Morgan's Laws, relations, equivalence relations, par-
titions, functions, countable and uncountable sets.

Number Theory:
Principle of mathematical induction, division algorithm, greatest common
divisor, divisibility, modular arithmetic.

Linear Algebra: Solution of a system of linear equations, Gaussian elimination, vector spaces,
subspaces, linear independence, basis, dimension, linear functionals, dual spaces, dual basis,
linear transformation, matrix of a linear transformation, change of basis, rank-nullity theorem,
trace, determinant, characteristic and minimal polynomial of a linear transformation, Cayley-
Hamilton theorem, eigenvalues, eigenvectors, special matrices (orthogonal, unitary, normal,
self-adjoint and hermitian) and their properties, diagonalizability, inner product spaces, Gram-
Schmidt process.

Complex variables: Algebra of complex numbers (addition, subtraction, multiplication, divi-
sion and conjugation), polar form of a complex number, De Moivre's formula, roots of complex
numbers, continuity and di_erentiability of a function of a single complex variable, analytic
functions, Cauchy-Riemann equations, Harmonic functions, trigonometric functions, exponen-
tial and complex logarithm.

Groups and Rings: Groups, subgroups, cyclic groups, permutation groups, homomorphisms,
isomorphisms, Lagrange's theorem, normal subgroups, quotient groups, isomorphism theorems,
rings, subrings, ideals, quotient rings, polynomial rings, integral domain, prime, principle and
maximal ideals, _elds, ring homomorphisms, ring isomorphisms.
Ordinary Di_erential Equations: First order di_erential equations (homogeneous, exact,
integrating factors, linear), solutions of second order equations with constant coe_cients.
Calculus & Real analysis: Real number system, supremum and in_mum, limits, continuity,
intermediate value theorem, extreme value theorem, di_erentiability, implicit di_erentiation,
Rolle's theorem, mean value theorem, maxima and minima, curve sketching, L'Hospital's rule,
sequences, subsequences, Bolzano-Weierstrass theorem, series, tests for convergence and diver-
gence, absolute convergence, Maclaurin series, Taylor series, Riemann integral, fundamental
theorem of calculus, improper integrals.

Vector Calculus: Vectors in R2 and R3, dot product, cross product, scalar triple product,
cartesian and vector equations of straight lines and planes, vector valued functions, paramet-
ric curves, functions of two and three real variables, continuity, partial derivatives, directional
derivatives, total derivatives, maxima and minima, saddle points, method of Lagrange multi-
plier, divergence, curl, gradient, polar, spherical and cylindrical coordinates, arc length, line
integrals, surface integrals, volume integrals, Green's, Stokes, and Divergence theorems.

Will you please provide the Indian Institutes of Science Education and Research , Bhopal Ph.D Maths Entrance Exam Syllabus ?

[Quick Sam]

As you are looking for the Indian Institutes of Science Education and Research , Bhopal Ph.D Maths Entrance Exam Syllabus , here I am provIding same for you
IISER Ph.D Maths Entrance Exam Syllabus
SYLLABUS FOR INTEGRATED Ph.D. MATHEMATICS ENTRANCE EXAMINATION
Set Theory: Sets, operations on sets, De Morgan's Laws, relations, equivalence relations, par-
titions, functions, countable and uncountable sets.
Number Theory: Principle of mathematical induction, division algorithm, greatest common
divisor, divisibility, modular arithmetic.
Linear Algebra: Solution of a system of linear equations, Gaussian elimination, vector spaces,
subspaces, linear independence, basis, dimension, linear functionals, dual spaces, dual basis,
linear transformation, matrix of a linear transformation, change of basis, rank-nullity theorem,
trace, determinant, characteristic and minimal polynomial of a linear transformation, Cayley-
Hamilton theorem, eigenvalues, eigenvectors, special matrices (orthogonal, unitary, normal,
self-adjoint and hermitian) and their properties, diagonalizability, inner product spaces, Gram-
Schmidt process.
Complex variables: Algebra of complex numbers (addition, subtraction, multiplication, divi-
sion and conjugation), polar form of a complex number, De Moivre's formula, roots of complex
numbers, continuity and dierentiability of a function of a single complex variable, analytic
functions, Cauchy-Riemann equations, Harmonic functions, trigonometric functions, exponen-
tial and complex logarithm.
Groups and Rings: Groups, subgroups, cyclic groups, permutation groups, homomorphisms,
isomorphisms, Lagrange's theorem, normal subgroups, quotient groups, isomorphism theorems,
rings, subrings, ideals, quotient rings, polynomial rings, integral domain, prime, principle and
maximal ideals, elds, ring homomorphisms, ring isomorphisms.
Ordinary Dierential Equations: First order dierential equations (homogeneous, exact,
integrating factors, linear), solutions of second order equations with constant coecients.
Calculus & Real analysis: Real number system, supremum and inmum, limits, continuity,
intermediate value theorem, extreme value theorem, dierentiability, implicit dierentiation,
Rolle's theorem, mean value theorem, maxima and minima, curve sketching, L'Hospital's rule,
sequences, subsequences, Bolzano-Weierstrass theorem, series, tests for convergence and diver-
gence, absolute convergence, Maclaurin series, Taylor series, Riemann integral, fundamental
theorem of calculus, improper integrals.
Vector Calculus: Vectors in R2 and R3, dot product, cross product, scalar triple product,
cartesian and vector equations of straight lines and planes, vector valued functions, paramet-
ric curves, functions of two and three real variables, continuity, partial derivatives, directional
derivatives, total derivatives, maxima and minima, saddle points, method of Lagrange multi-
plier, divergence, curl, gradient, polar, spherical and cylindrical coordinates, arc length, line
integrals, surface integrals, volume integrals, Green's, Stokes, and Divergence theorems.