### Syllabus for the Preliminary Examination in Applied Mathematics

It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.

__The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:__

**1. Banach spaces:**

Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; Hahn-Banach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and Ascoli-Arzelà Theorem; compact operators and the Fredholm alternative.

**2. Hilbert spaces:** Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.

**3. Distributions:** Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.

__These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:__

**4. The Fourier Transform and Sobolev Spaces:** The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.

**5. Variational Boundary Value Problems (BVP):** Weak solutions to elliptic BVP’s; variational forms; Lax-Milgram Theorem; Green’s functions.

**6. Differential Calculus in Banach Spaces and Calculus of Variations:** The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.

**References: **

The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.

1. C. Carath'eodory, *Calculus of Variations and Partial Differential Equations of the First Order,* 2nd English Edition, Chelsea, 1982.

2. F.W.J. Olver, *Asymptotics and Special Functions,* Academic Press, 1974.

3. M. Reed and B. Simon, *Methods of Modern Physics,* Vol. 1, Functional analysis.

4. R.E. Showalter, *Hilbert Space Methods for Partial Differential Equations,* available at World Wide Web address http://ejde.math.swt.edu//mono-toc.html .

5. A. Avez, *Introduction to Functional Analysis,* *Banach Spaces, and Differential Calculus,* Wiley, 1986.

6. L. Debnath and P. Mikusi'nski, *Introduction to Hilbert Spaces with Applications,* Academic Press, 1990.

7. I.M. Gelfand and S.V. Fomin, *Calculus of Variations,* Prentice-Hall, 1963.

8. E. Kreyszig, *Introductory Functional Analysis with Applications,* 1978.

9. J.T. Oden and L.F. Demkowicz, *Applied Functional Analysis,* CRC Press, 1996.

10. W. Rudin, *Functional Analysis,* McGraw-Hill, 1991.

11. W. Rudin, *Real and Complex Analysis,* 3rd Edition, McGraw-Hill, 1987.

12. K. Yosida, *Functional Analysis,* Springer-Verlag, 1980.