Penn Arts & Sciences Logo

Written Preliminary Exam

The Written Preliminary Exam is taken by all incoming AMCS graduate students at the University of Pennsylvania, just prior to the start of the fall semester (generally in late August). It plays three roles:

  • It serves as a placement exam, to help determine whether students should begin with 500-level courses or with 600-level courses (or with a mixture).
  • It is a requirement for each of the graduate degrees in applied mathematics, in order to ensure that those who receive graduate degrees have a solid mathematical foundation.
  • It provides an incentive for incoming grad students to review basic material, which will then help them in their beginning graduate classes.

PhD students who do not pass the exam on their first attempt will have one more  chance to pass it at the end of the spring semester (generally in late April or early May).  PhD students who do not pass the exam by the end of their first year will be asked to leave the program.

Masters students who do not pass the exam on their first attempt will have two more opportunities to pass this exam. Students who fail the exam are required to retake the exam the next time it is offered. If a student does not do so, then they will be given a score of zero, and loose that opportunity to retake exam. While we strongly encourage masters students to prepare before their arrival, with the intention of passing the exam on their first try, this first attempt serves primarily as a placement exam.

All students who do not pass the Written Prelim on the first try are strongly encoraged to take the Proseminar (MATH 504, 505), which helps to prepare you for this exam. You will also be directed to other 400/500-level math courses during your first year, to strengthen your problem solving ability, and background in mathematics.

The written preliminary exam focuses on the material from an undergraduate mathematics program that is most important to those entering a applied mathematics graduate program. The exam is given in two 2.5 hour sessions, either on a single day, or on consecutive days. Each  parts consists of 6-8 problems.

The exam contains problems in linear algebra, advanced calculus, basic complex analysis and probability. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples. Each part of the exam constains a mixture of types of problems, and a mixture of subjects.

The key to success on the preliminary exams is practice!


Books that cover the material at an appropriate level are as follows.  We list more than one book in some areas only to maximize the probability that the list contains a book you are familiar with.

Linear algebra: Strang, Gilbert "Linear Algebra and its Applications"

Real analysis: Rudin, Walter "Principles of Mathemaical Analysis" or Strichartz, Robert "The Way of Analysis"

Probability theory: Ross, Sheldon "A First Course in Probability" or Hoel, Port and Stone "Introduction to Probability Theory"

Complex Analysis: Conway, John B. "Functions of One Complex Variable" or Bak and Newman "Complex Analysis" or Alfors, Lars "Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable"


Here are some practice problems and previous years' exams:


You may find solutions to some of these problems online or in books, but reading other people's solutions is a very poor way to study. Hard work and diligence are a much better bet, so try to solve as many of these problems as you can on your own.

The following list of topics gives a general idea of the material that is covered on the exam:

  • I. Analysis
    • Continuity, uniform continuity, properties of real numbers, intermediate value theorem, metric spaces, topological spaces, compactness, epsilon-delta proofs.
    • Differentiable functions of one variable: differentiation, Riemann integration, fundamental theorem of calculus, mean-value theorem, Taylor's theorem
    • Sequences and series of numbers and functions, uniform convergence, equicontinuity, interchange of limit operations, continuity of limiting functions.
    • Ordinary differential equations (separable, exact, first order linear, second order linear with constant coefficients), applications such as orthogonal trajectories.
    • Multivariable calculus: partial derivatives, multiple integrals, integrals in various coordinate systems, vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem, divergence theorem and Stokes' theorem), inverse and implicit function theorems, Lagrange multipliers.
    • Power series and contour integration.
    • Basics of Fourier series.
  • II. Linear Algebra
    • Linear Algebra:
      • Vector spaces over RC, and other fields: subspaces, linear independence, basis and dimension.
      • Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.
      • Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant.
      • Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.
      • Positive definite matrices and the variational characterization of eigenvalues and eigenspaces.
    • Numerical linear algebra
      • Basic algorithms for solving linear systems of equations
      • Notions of stability and conditioning
      • Basic algorithms for finding eigenvalues and eigenvectors 
  • III. Probability and statistics
    • The basic notions  of events and probability, simple discrete distributions
    • Independence of events
    • Random variables
    • Moments, the characteristic function
    • Simple examples of estimators, and the notion of bias
  • IV. Complex analysis
    • Definitions of analytic functions
    • Cauchy theorem and integral formula
    • Power series
    • Residue calculations
    • Elementary conformal maps