Math 537
Principles of Applied Mathematics
Spring 2017

Course Info

Overview

We will cover finite and infinite dimensional vector spaces, spectral theory of differential operators, distributions and Green’s functions applied to initial and boundary value problems. Potential theory, and conformal mappings. Asymptotic methods and perturbation theory. This will be roughly chapters 1-4, 6-7, 10 from the book.

Grading

Text Book

James P. Keener, Principles Of Applied Mathematics: Transformation And Approximation, Second Edition, Westview Press 2000

Software

I will use Mathematica throughout the course to demonstrate some key concepts. You will also be asked to use this or similar software, such as WolframAlpha, Maple, MATLAB (through the symbolic toolbox), or Chebfun, on the homework assignments. The purpose here is to help you build a qualitative understanding of the material and to aid you with some tedious computations for the homework problems. All of this software is available in most computing labs around the university, including the mathematics computing lab. WolframAlpha is available through most web browsers and there is even an app for it for your phones or tablets.

Lectures

Topics that will be covered in a particular lecture will appear a day or two before the lecture. I expect you to have read the relavent sections in the book before the lecture and to be prepared to answer discuss the material.
Day Topics Notes
09-Jan Section 1.1: finite dimensional vector spaces
11-Jan Section 1.2: Spectral theory for matrices
Section 1.3: Geometrical significance of eigenvalues
18-Jan Section 1.4: Fredholm Alternative
Section 1.5: Least Squares Solutions-Pseudo Inverses
23-Jan Section 1.5: Singular value decomposition
Section 1.6: Applications of eigenvectors and values
25-Jan Section 2.1: Complete vector spaces
Section 2.2: Approximation in Hilbert spaces
30-Jan Section 2.2: Approximation in Hilbert spaces: Fourier series and Orthogonal Polynomials
1-Feb Section 2.2: Approximation in Hilbert spaces: Discrete Fourier Series and the FFT
6-Feb Section 2.2: Approximation in Hilbert spaces: Wavelets
8-Feb Section 2.2: Approximation in Hilbert spaces: Multiresolution analysis and Finite Elements
13-Feb Section 2.2: Approximation in Hilbert spaces: Finite Elements
Section 3.1: Integral equations
Section 3.2: Bounded linear operators
15-Feb Section 3.2: Bounded linear operators
20-Feb Section 3.3: Compact linear operators
22-Feb Section 3.3: Compact linear operators
27-Feb Section 3.4: Spectral theory for compact operators
1-Mar Section 3.5: Resolvent and pseudo-resolvent kernels
6-Mar Section 3.5: Resolvent and pseudo-resolvent kernels
Section 3.6: Approximate solutions
8-Mar Section 4.1: Delta functions
13-Mar Section 4.1: Delta functions Section 4.2: Green's functions
15-Mar Section 4.2: Green's functions
3-Apr Section 4.3: Differential operators
5-Apr Section 4.3.2: Adjoints of Differential operators
10-Apr Section 4.3.4: Fredholm Alternative for differential equations
Section 4.4: Least squares solutions
12-Apr Section 4.5: Eigenfunction expansions
17-Apr Section 4.5: Eigenfunction expansions
Section 4.5.2: Orthogonal polynomials
19-Apr Section 6.1: Complex valued functions
Section 6.2: The calculus of complex functions

Homework

Your submitted homework should show all necessary work you used to solve the problems. Mathematical statements should be complete (or nearly complete) sentences and the reasoning and logic underlying all arguments should be clearly spelled out. Any work in Mathematica (or other software) should be noted and included in a readable format (e.g., only the necessary work with concise codes). Do not include pages and pages of printouts from the computer of results that are not important to the problem at hand. Homework assignments will mostly be of the analytical nature, but there may be some computational-type problems.

Format

The ideal way to write up your homework assignments is to use LaTeX, but legible, hand-written assignments are also acceptable.

You are also encouraged to check out these suggestions from the Mathematics Department at Harvey Mudd on formatting homework assignments in mathematics.

Assignments

Due date Problem set Notes
23-Jan 1.1: 1, 2, 7, 9a, 10 (also plot all 5 of the polynomials over [-1 1])
1.2: 1, 2a, 2b, 3, 4, 6a, 6d, 10a
1.3: 3
1-Feb 1.4: 1, 4
1.5: 1bc, 2, 5, 14
2.1: 2, 3, 4
24-Feb 2.2: 1, 2b, 2c, 7, 8, 9, 14, 17, 20, 22, 25a
15-Mar 3.1: 1
3.2: 2, 3
3.3: 1
3.4: 1, 2b, 3, 6
3.5: 1b, 2b
3.6: 4, 6
5-Apr 4.1: 2, 5, 9, 11, 12
4.2: 1, 3, 6, 9, 13
1-May 4.3: 1, 3, 6
4.4: 3, 7
4.5: 3, 6
6.1: 2
6.2: 4, 6, 12

References

The following are some other good references for the material covered in this course:

Please e-mail me regarding any problems with the links on this page.