University Of Delhi Syllabus for Ph.D. Entrance Test

I want to take admission in the University Of Delhi in the Ph.D course? so I am searching for the Ph.D. Entrance Test syllabus? Can you please tell me from where I can download the Ph.D. Entrance Test syllabus of University Of Delhi?

[Kiran Chandar]

You are asking for the University Of Delhi Ph.D. Entrance Test syllabus. Here I am uploading a file that contains the Ph.D. Entrance Test syllabus of University Of Delhi. You can download it from here. Here I am also providing you some content of the file this is as follows:

Section I – Analysis:
Finite, countable and uncountable sets, bounded and unbounded sets, Archimedean property,
ordered field, completeness of ℝ, sequence and series of functions, uniform convergence,
Riemann integrable functions, improper integrals, their convergence and uniform convergence,
Fourier series. Partial and directional derivatives, Taylor’s series, implicit function theorem, line
and surface integrals, Green’s theorem, Stoke’s theorem.
Elements of metric spaces, convergence, continuity, compactness, connectedness,
Weierstrass’s approximation theorem, completeness, Baire’s category theorem, Bolzano-
Weirstrass theorem, compact subsets ofℝ_, Heine-Borel theorem,
Lebesgue outer measure, Lebsegue measure and Lebsegue integration, Riemann and
Lebesgue integrals.
Complex numbers, analytic functions, Cauchy-Riemann equations, Riemann sphere and
stereographic projection, lines, circles, crossratio, Mobius transformations, line integrals,
Cauchy’s theorems, Cauchy’s theorem for convex regions, Morera’s theorem, Liouville’s
theorem, Cauchy’s integral formula, zero-sets of analytic functions, exponential, sine and cosine
functions, power series representation, classification of singularities, conformal mapping,
contour integration, fundamental theorem of algebra.
Banach spaces, Hahn-Banach theortem, open mapping and closed graph theorem, principle of
uniform boundedness, boundedness and continuity of linear transformations, dual spaces,
embedding in the second dual, Hilbert spaces, projections, orthonormal bases, Riesz
representation theorem, Bessel’s inequality, Parseval’s identity.
Elements of Topological spaces, continuity, convergence, homeomorphism, compactness,
connectedness, separation axioms, first and second countability, separability, subspaces, product
spaces.


Section II – Algebra:
Space of n-vectors, linear dependence, basis, linear transformations, algebra of matrices, rank
of a matrix, determinants, linear equations, characteristic roots and vectors.
Vector spaces, subspaces, quotient spaces, linear dependence, basis, dimension, the algebra of
linear transformations, kernel, range, isomorphism,linear functional, dual space, matrix
representation of a linear transformation, change of bases, reduction of matrices to canonical
forms, inner product spaces, orthogonality, eigenvalues and eigenvectors, projections, triangular
form, Jordan form, quadratic forms, reduction of quadratic forms.
Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups,
permutation groups, Cayley’s theorem, Symmetric groups, alternating groups, simple groups.
conjugate elements and class equations of finite groups, Sylow’s theorem, solvable groups,
Jordan-Holder theorem, direct products, structure theorem for finite abelian groups.
Rings, Ideals, prime and maximal ideals, quotient ring, integral domains, Euclidean domains,
principal ideal domains, unique factorization domains,polynomial rings, chain conditions on
rings,fields, quotient fields, finite fields, characteristic of field, field extensions, elements of
Galois theory, solvability by radicals, ruler and compass construction.


Section III- Differential Equations and Mechanics:
First order ODE, singular solutions, initial value problems of first order ODE, general theory of
homogeneous and non-homogeneous linear ODEs, variation of parameters, Lagrange’s and
Charpit’s methods of solving first order PDEs, PDEs of higher order with constant coefficients.
Existence and uniqueness of solution ( , ) dy
dx f x y , Green’s function, Sturm-Liouville boundary
value problems, Cauchy problems and characteristics, classification of second order PDE,
separation of variables for heat equation, wave equation and Laplace equation,
Equation of continuity in fluid motion, Euler’s equations of motion for perfect fluids, two
dimensional motion, complex potential, motion of sphere in perfect liquid and motion of liquid
past a sphere, vorticity, Navier-Stoke’s equations of motion for viscous flows, some exact
solutions.