# PhD Qualifying Exam

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#### 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.

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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

**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