---------------------------------------------------------------- MATH 580 - Fall 2003 Introduction to Applied Mathematics I Instructor: Andrew Belmonte Email: belmonte@math.psu.edu Time: TR 9:45 - 11:00 AM Location: 102 McAllister Bldg Schedule Number: 296764 (previously Math 597K) This graduate course will cover some of the basic techniques in applied mathematics that are not necessarily included in regular mathematics courses but are essential to the fields of physics, chemistry, engineering, computer science, and other sciences. Its purpose is to introduce graduate students with diverse backgrounds to several of the fundamental ideas in applied mathematics, in order to prepare them for other graduate courses that involve the use of mathematical or computational techniques. This two semester course is the result of a collaborative effort between several departments at Penn State. During the first semester an emphasis will be placed on the physical and mathematical notions of vectors and tensors, and also on the following unifying concepts: projection/decomposition, transforms/mappings/operators, and bifurcations/linear stability. SYLLABUS Vectors \& Tensors: linear vector spaces; completeness; review of eigenvalues and matrices; matrices and tensors; calculus of vectors and tensors; coordinate transformations. Applied Functional Analysis: function spaces (Banach \& Hilbert); Riesz representation; Fredholm alternative; distributions; operators; compactness. Linear Transforms \& Spectral Theory: orthogonal functions; Fourier and Laplace transforms; Nyquist \& sampling; applications: power spectra, FFT, wavelets. Differential Equations (Ordinary \& Partial): fundamental solutions; stability \& instability; bifurcations; Sturm-Liouville; linear stability \& normal mode analysis; similarity solutions. Complex Variables: analytic functions; Cauchy's theorem; conformal mapping, with applications. TEXTS: J. P. Keener, Principles of Applied Mathematics, 2nd ed. (Perseus Books, 2000). R. Aris, Vectors, Tensor, and the Basic Equations of Fluid Mechanics (Dover, 1962). ----------------------------------------------------------------