PhD Qualifying Exam



Probability, linear algebra, advanced calculus, ordinary and partial differential equations, real analysis, complex variables (including contour integration).  The material will be tested at the advanced undergraduate level.


There will be a 3-hour morning session (9:00AM-12:00PM), and a 3 hour afternoon session (2:00PM-5:00PM). Both sessions will contain both elementary and challenging problems from each of the coverage topics.


The exam is closed-book, closed-notes; paper will be provided. No calculators or other books, papers, notes, backpacks, etc. will be permitted in the room. The exam will be strictly supervised.

Covered Material with Suggested Reading:

The following material is selected to exemplify the breadth and depth of topics covered on the qualifying exam.



S.M. Ross
, A First Course in Probability

Chapter 1:  Combinatorial analysis
Chapter 2:  Axioms of probability
Chapter 3:  Conditional probability and independence
Chapter 4:  Random variables
Chapter 5:  Continuous random variables
Chapter 6:  Jointly distributed random variables
Chapter 7:  Properties of expectation
Chapter 8:  Limit theorems


Linear Algebra

Carl D. Meyer
, Matrix Analysis and Applied Linear Algebra

Chapter 1:  Linear equations
Chapter 2:  Rectangular systems
Chapter 3:  Matrix algebra
Chapter 4:  Vector spaces
Chapter 5:  Norms, inner products and orthogonality
Chapter 6:  Determinants
Chapter 7:  Eigenvalues and eigenvectors


Ordinary and Partial Differential Equations


K. Nagle, E. Saff and A. Snider, Fundamentals of Differential Equations

Chapter 2:  First order differential equations
Chapter 4:  Linear second order differential equations
Chapter 5:  Introduction to systems and phase plane analysis
Chapter 6:  Theory of higher order linear differential equations
Chapter 7:  Laplace transforms
Chapter 8:  Series solutions of differential equations
Chapter 13:  Existence and uniqueness theory

Walter Strauss, Partial Differential Equations

Chapter 1:  First and second order PDEs.
Chapter 2:  Waves and diffusions
Chapter 3:  Reflections and sources
Chapter 4:  Boundary problems

J. Brown, R. Churchill, Fourier Series and Boundary Value Problems

Chapter 1:  Fourier series
Chapter 2:  Convergence of Fourier series
Chapter 4:  The Fourier method
Chapter 5:  Boundary value problems
Chapter 6:  Fourier integrals and applications
Chapter 7:  Orthonormal sets
Chapter 8:  Sturm-Liouville problems


Real Analysis

W. Rudin, Principles of Mathematical Analysis

Chapter 1:  The real and complex number systems
Chapter 2:  Basic Topology
Chapter 3:  Numerical sequences and series
Chapter 4:  Continuity
Chapter 5:  Differentiation
Chapter 6:  The Riemann – Stieltjes  integral
Chapter 7:  Sequences and series of functions
Chapter 8:  Some special functions
Chapter 9:  Functions of several variables

Erwin Kreyszig, Advanced Engineering Mathematics

PART B Linear algebra and vector calculus
Chapter 9:  Vector differential calculus. Grad. Div. Curl
Chapter 10:  Vector integral calculus. Integral theorems and applications


Complex Variables

J. Brown, R. Churchill, Complex Variables and Applications

Chapter 1:  Complex numbers
Chapter 2:  Analytic functions
Chapter 3:  Elementary functions
Chapter 4:  Integrals
Chapter 5:  Series
Chapter 6:  Residues and poles
Chapter 7:  Applications of residues
Chapter 8:  Mapping by elementary functions
Chapter 9:  Conformal mapping