Math 427/527
Introduction to Applied Mathematics for Scientists and Engineers
Fall 2016

Course Info

Overview

We will cover vector integral calculus, Fourier series and transforms, series solutions to differential equations, Sturm-Liouville problems, wave equation, heat equation, Poisson equation, analytic functions, and contour integration. This will be Chapters 5, 10, 11-14, and 17 (time-permitting).

Please see the syllabus for further details about the course.

Text Book

Erwin Kreyszig, Advanced Engineering Mathematics, Tenth Edition, Wiley 2011

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 questions about the material.
Day Topics Notes
22-Aug-2016
Section 5.1: Power series method for variable coefficient ODEs
Please also review Sections 1.1-3.3 (Review of ordinary differential equations (ODES))
24-Aug-2016 Section 5.2: Legendre's equation and Legendre polynomials
29-Aug-2016 Section 5.2: Legendre's equation and Legendre polynomials
Section 5.3: Extended power series method: Frobenius method
Lecture notes
31-Aug-2016 Section 5.4: Bessel's equation, and Bessel functions of the first kind Lecture notes
07-Sep-2016 Section 10.1: Line integrals
Section 10.2: Path independence of line integrals
12-Sep-2016 No Class, but read section 10.3
14-Sep-2016 Section 10.2: Path independence of line integrals
Section 10.3: Double integrals
19-Sep-2016 Section 10.4: Green's Theorem
Section 10.5: Surfaces
21-Sep-2016 Section 10.6: Surface integrals
Section 10.7: Triple integrals and Gauss's divergence theorem
26-Sep-2016 Section 10.7: Triple integrals and Gauss's divergence theorem
Section 10.8: Applications of the divergence theorem
28-Sep-2016 Section 10.8: Applications of the divergence theorem
Section 10.9: Stokes's theorem
03-Oct-2016 Section 10.9: Stokes's theorem
05-Oct-2016 Section 11.1: Fourier Series
10-Oct-2016 Review for midterm exam
12-Oct-2016 Midterm exam - Chapters 5 and 10
17-Oct-2016 Section 11.2: Fourier series over arbitrary domains, etc.
Section 11.3: Forced oscillations
19-Oct-2016 Section 11.3: Forced oscillations
Section 11.4: Approximation by trigonometric polynomials
24-Oct-2016 Section 11.5: Sturm-Liouville problems
Section 11.6: Orthogonal series
26-Oct-2016 Section 11.6: Orthogonal series
Mathematica codes: FourierLegendreSeries.nb, FourierBesselSeries.nb.
Matlab codes (requires Chebfun): FourierLegendreSeries.m, FourierBesselSeries.m.
31-Oct-2016 Section 11.7: Fourier integral
Section 11.8: Fourier cosine and sine transforms
2-Nov-2016 Section 11.8: Fourier cosine and sine transforms
Section 11.9: Fourier transform
Ch. 11 Lecture notes
7-Nov-2016 Section 11.9: Convolution
Section 12.1: Intro to PDEs
Section 12.2: Derivation of the wave equation
9-Nov-2016 Section 12.3: Solution of the wave equation
Wave equation example
14-Nov-2016 Handout take-home midterm
Section 12.4: D'Alembert's solution to the wave equations
D'Alembert example
16-Nov-2016 Take-home midterm due
Section 12.6: Heat equation: solution by Fourier series
Heat equation example
28-Nov-2016 Section 12.6: Heat equation: solution by Fourier series
Heat equation example
30-Nov-2016 Section 12.7: Heat equation: Fourier transform solution
Section 12.12: Heat equation: Laplace transform solution
05-Dec-2016 Section 12.10: Wave equation and Laplace's equation on the disk
07-Dec-2016 Section 12.10: Wave equation and Laplace's equation on the disk
Review for final exam
Harmonics of a circular drum

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 involve a mix of analytical and computational work.

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
07-SEP-2016 HW01 Select solutions
21-SEP-2016 HW02 Select solutions
05-OCT-2016 HW03 Select solutions
26-OCT-2016 HW04 Select solutions
09-NOV-2016 HW05 Select solutions
30-NOV-2016 HW06 Select solutions
09-DEC-2016 HW07 Select solutions

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.