PhD Qualifying Exam

 

Coverage:

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.


Format:

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.


Rules:

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.

 

Probability


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


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

 

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

 

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