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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).
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