Department of Mathematics
Math 189B. Topics in Applied Mathematics Fall
2019
Time and Place: MWF 9:00 am - 9:50 am Millikan 2393.
Instructor: Dr.
Adolfo J. Rumbos
Office: Andrew 2287.
Phone/e-mail: ext. 18713 /
Office Hours: TuTh 9:00 am - 10:00 am, or by appointment
Course Notes: http://pages.pomona.edu/~ajr04747/
Prerequisites: Multivariable Calculus, Linear Algebra
and Differential Equations (Math 102 at
Course
Description. The topic for this year is Variational Methods and Optimization. This course is an introduction to the calculus of variations and
the variational approach in the theory of
differential equations. The calculus of variations is a subject as old as the
Calculus of Newton and Leibniz. It arose
out of the necessity of looking at physical problems in which an optimal
solution is sought; e.g., which configurations of molecules, or paths of
particles, will minimize a physical quantity like the energy or the action? Problems like these are known as variational problems.
From its
beginnings, the calculus of variations has been intimately connected with the
theory of differential equations; in particular, the theory of boundary value
problems. Sometimes a variational problem leads to a differential equation that
can be solved, and this gives the desired optimal solution. On the other hand, variational
methods can be successfully used to find solutions of otherwise intractable
problems in nonlinear partial differential equations. This interplay between the theory of
differential equations and the calculus of variations will be one of the major
themes in the course.
Course Requirements.
Reading assignments will be given according to the attached (tentative)
schedule. Problem sets will be assigned
and collected on a weekly basis.
Students are strongly encouraged to work on every assigned problem. Students will also be expected to give a
written and oral formal presentation on some topic of their interest related to
the course material, or some suggested one (see attached list of special
topics).
Grading Policy. Grades will be based on the homework, two
examinations (see attached schedule), plus a written and oral
presentation. The grades will be
computed as follows:
Homework 20%
Two
exams 50%
Paper
and presentation 30%
Math 189B Fall
2019
Tentative Schedule of Topics,
Presentations and Examinations
Date Topic
W Sep.
4 Soap films and minimal surfaces.
F Sep. 6 Variational problems
M Sep. 9 Variational problems (continued)
W Sep. 11 Normed linear spaces
F Sep.
13 Continuous functionals on normed linear spaces
M Sep. 16 Indirect Methods
W Sep. 18 Gateaux derivatives and the first variation
F Sep.
20 The Euler-Lagrange
equations
M Sep. 23 The Euler-Lagrange equations (continued).
W Sep.
25 Examples
F Sep.
27 Problems
M Sep. 30 Convex
functionals.
W Oct.
2 Convex functionals (continued).
F Oct.
4 Minimization of convex functions
M Oct. 7 Convex minimization theorem
W Oct.
9 Review
F Oct.
11 Exam 1
M Oct. 14 Isoperimetric
problems
W Oct.
16 Isoperimetric problems
(continued)
F Oct. 18 Examples
M Oct.
21 Fall Recess!
W Oct.
23 Direct methods
F Oct.
25 Direct methods
(continued)
M Oct. 28 The
variational approach
W Oct.
30 Hilbert space methods
F Nov.
1 The Dirichlet
principle
M Nov. 4 Solving the Dirichlet
problem
W Nov. 6 Solving
the Dirichlet problem (continued)
F Nov.
8 Problems
Date Topic
M Nov.
11 Eigenvalues of the Laplacian
W Nov. 13 Sturm-Lioville problems
F Nov. 15 Problems
M Nov.
18 Examples
W Nov. 20 Review
F Nov. 22 Exam 2
M Nov.
25 Examples
W Nov. 27 Thanksgiving Recess!
F Nov. 29 Thanksgiving Recess!
M Dec. 2 Special Topic
W Dec. 4 Special Topic
F Dec. 6 Special
Topic
M Dec. 9 Special Topic
W Dec.
11 Special Topic