



MATH 597K Introduction
to Applied Mathematics
Course Purpose:
This is the first semester of a oneyear introductory graduate
course on applied mathematics.
This is the second time we offer this course.
The course is designed for graduate students who wish to learn some
basic mathematical techniques that can be applied in their research
and future career. Because of the rapid development of computer and
network technologies, more and more mathematical techniques are used
in more and more fields such as physics, chemistry, biology,
economics, finance, engineering, meteorology, rheology, etc..
This course will cover some of
the very basic mathematical techniques that are often not covered in
regular undergraduate mathematical courses but are essential to the
aforementioned application areas.
This course has been designed by a group of computational and
applied mathematicians from the department of mathematics with close
collaborations with many professors from different departments at Penn
State University. The syllabus of the course has been extensively
debated and carefully designed to address the special needs of
students with diversely different backgrounds. It is expected this
course will be prerequisite for other basic graduate courses in
computational and applied mathematics offered by the department of
mathematics at Penn State. It is also expected that students who have
taken this course sequence will have a solid foundation to take most
other graduate courses on campus that involve the use of mathematical
and computational techniques.
Introduction to Applied Mathematics I (Fall 2002)
Prerequisite: Math 405 (equivalent) or consent of the instructor
 Cartesian vectors and tensor calculus (2 weeks): vector
operations; Cartesian tensor operations; Kronecker delta and
alternating tensor; surface and line integrals; Green's and Stokes'
theorems; basic vector operations in spherical and cylindrical
coordinates.
 Complex variables (1.5 weeks): analytic functions; Cauchy's
theorem and integration formula; complex vectors.
 Applied functional analysis (2 weeks): Banach and Hilbert spaces; Riesz
representations; Fredholm alternative theorem; spectral theory for compact
operator
 Linear transforms (Fourier, Laplace) (1.5 week): Fourier and Laplace
integral and properties, Fourier series vs. Fourier integrals; SturmLiouville
operator.
 Ordinary differential equations (2 weeks): stability of first order linear
systems, perturbation methods. Special functions.
 Partial differential equations (5 weeks): basic properties of
solutions; separation of variables; Fourier and Laplace transform
solutions; infinite and semiinfinite domain solutions; similarity
solutions; special functions (Bessel, Legendre, spherical harmonics);
boundary value (eigenvalue) problems.
 Homogenizations (1 week)
Introduction to Applied Mathematics II (Spring 2003, 59xX)
Prerequisite:
Introduction to Applied Mathematics I or consent of
the instructor
 Calculus of variations (approximate 3 weeks): EulerLagrange
equations; constraint problems; Hamilton's principle; stability and
second variations; applications.
 Partial differential equations (approximate 3 weeks): Green's
functions and fundamental solutions; eigenfunction expansions and
Galerkin's method; Sobolev space and weak solutions.
 Asymptotic expansions (approximate 1 week): Laplace method; method
of steepest descents; method of stationary phase.
 Regular perturbation theory (approximate 2 weeks): oscillations
and periodic solutions; perturbation of eigenvalues; LyapunovSchmidt
method.
 Singular perturbation theory (Approximate 2 weeks): initial value problems;
boundary value problems.
 Wavelet analysis (approximate 1 week).
 Stochastic differential equations, BlackScholes model (approximate 2
weeks).
Test and Grades
There will be one midterm exam and one final exam per course. The final
course grade will be determined as follows:
30% homework+ 30% midterm exam + 40 % final exam.
Lecture Schedule: MWF 9:059:55AM, 123 Chambers Building
Instructor: Yuxi Zheng,
yzheng@math.psu.edu.
Course Texts
1. A.I. Borisenko and I. E. Tarapov, translated by R. A. Silverman,
Vector and Tensor Analysis with Applications, Dover. 1968.
2. J. Keener, Principles of Applied Mathematics: Transformation and
Approximation, Perseus Books, 2000.
Reference Books
1. E. C. Young, Vector and Tensor Analysis , Marcel Dekker, Inc.
New York, 1992.
2. J. G. Simmonds, A Brief on Tensor Analysis, Springer, 1982.
3. H. M. Schey, Div, Grad, Curl and All That: an Informal Text on
Vector Calculus, New York, W.W. Norton, 1997.
4. H. F. Weinberger, A First Course in Partial Differential
Equations with Complex Variables and Transform Methods, Dover.
5. I. M. Gel'fand, S. V. Fomin, Calculus of Variations,
PrenticeHall, 1963.
More to be given later in the class.
Course Oversight Committee
© 20012002 Center for Computational Mathematics and Applications
Last Updated August 30, 2002. Originally designed by Ludmil
Zikatanov and Jason Nichols

