• 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.

The preliminary exam focuses on the key material from an undergraduate mathematics program that is most important to those entering a applied mathematics graduate program. The first half of the exam is given in the morning, and the second half in the afternoon. Each of these two parts consists of six problems, and students are given two and a half hours for each part.

The exam consists of 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 (morning and afternoon) constains a mixture of types of problems, and a mixture of analysis and algebra problems.

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 R, C, 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
  

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