# ACM95b/ACM100b

## Introductory Methods of Applied Mathematics

Topics,Texts, Handouts, Homework, Exams, Grading, Ombudspersons, Schedule, Sections, TAs

### Instructor

Sterl Phinney
125 Bridge Annex
esp [at] tapir [dot] caltech [dot] edu

### Course web pages:

http://www.its.caltech.edu/~esp/acm95b/ACM95b.html
(rules, syllabus, handouts, problem sets and solutions)

http://underground.acm.caltech.edu/

### Overview

The second term covers ordinary differential equations, transforms and special functions.

### Topics

Introduction to ordinary differential equations, linear first order ODEs, integrating factors, integral curves, singular points, series solution, convergence, existence and uniqueness, the view in the complex plane, nonlinear first order ODEs, Picard's existence and uniqueness theorem, introduction to numerical methods, explicit and implicit Euler methods, trapezoidal rule, explicit Runge-Kutta methods, truncation error, order of accuracy, solution error, stiff ODEs,  second order linear IVPs, reduction of order, variation of parameters, the Laplace transform, convolution, initial value problems with discontinuous and impulsive forcing functions, inverting Laplace transforms with the Mellin inversion formula and the Bromwich contour, higher order linear IVPs, conversion to first order systems, the fundamental matrix, eigenvalues, eigenvectors and generalized eigenvectors, decoupling systems via similarity transformations, phase plane interpretations, nonlinear first order ODEs, linear stability analysis, Poincare-Bendixson theorem, linear equations with analytic coefficients, series solutions near ordinary points, solution behavior near singular points, regular singular points, Euler equations, solutions near regular singular points by the method of Frobenius, asymptotics and WKBJ solutions, separation of variables for partial differential equations, boundary value problems, Fourier series, Gibb's phenomenon, Fourier transforms, the delta function, Bessel functions and other special functions, Green's functions, eigenvalue problems, eigenfunction expansions, Sturm-Liouville theory.

### Suggested Texts

There is no ideal text which covers the material of this course. I suggest you get at least one of the following three recommended texts (available in the ACM 95 section of the bookstore) which is most compatible with your learning style and future plans.
1. G. F. Carrier and C. E. Pearson Ordinary Differential Equations, SIAM 1991. This is my model of an ideal textbook: it is concise but offers very deep insights into the important issues. However, you have to work for it, for it teaches by the Socratic method. It guides you through a carefully selected set of questions and problems so that you `discover' the theory of ODEs for yourself. It covers much of this term's material, but not all (in particular, no complex variables -e.g. inverse Laplace transforms, convergence and extension of solutions, some of the special functions). This is not the book for cramming recipes the night before midterms. That would be the next book:
2. G. Arfken Mathematical Methods for Physicists, 5th edition Harcourt/Academic. This is an encyclopedic reference book masquerading as a textbook. It covers almost everything in this course (and a lot more: e.g. complex analysis, tensors, group theory, but not much nonlinear ODEs), in a rather cookbook style. The reader of all of this book would have more techniques at his/her disposal than the reader of Carrier and Pearson, but be less likely to know when to use them, when they might fail, when something had gone wrong or what to do when confronted by something really novel. Nevertheless, if you get this book, you will refer to it with profit many, many times in the coming decades. Despite the title, this is especially true if you plan to be an engineer (in my view, engineers are physicists who solve the problems that physicists abandoned because they were too hard: e.g. design a real loudspeaker instead of a microscopic spherical one with infinite inertia driven by an infinite force).
3. S. Hassani Mathematical Physics, Springer1998. The intersection of this book with the material of this course is somewhat smaller than it is for the previous two. However it covers especially well and much more thoroughly those portions of the material (and lots more besides) most relevant for theoretical physicists of the quantum mechanical and field theoretic persuasions.

### Lecture Handouts

Leftover paper copies of handouts are placed on the ACM95b shelf in 124 Bridge. PDF versions are posted here (read with open source xpdf , or closed source Adobe Acrobat reader).

Jan 5 Introduction to Differential Equations
Jan 16 Method of Frobenius
Jan 21 Laplace Transforms, Table of Laplace Transforms
Jan 28 Delay differential equation
Feb 18 Fourier solution of heat equation
Mar 1 Fourier Transforms
Mar 3 Green's functions for 2nd order LODEs
Mar 10 Rayleigh-Ritz and the Finite Difference Solution to Sturm-Liouville problems

### Problem Sets

See links here for actual problem sets and solutions.

If you were in ACM95a/100a last term, please use the same grading section number you used last term. If you are new this term,  please register yourself on the underground, and record the
grading section it assigns you. PLEASE write this underground grading section number (NOT the registrar-assigned section number) on every problem set and exam you hand in.

Problem sets posted online at 3:30pm (generally Fridays)
Due at 3pm in the slot of Firestone 303 (generally one week after posting)

Collaboration policy: Before starting the homework sets, you are encouraged to review the lecture notes and handouts, and read the relevant sections of the textbook(s) of your choice (not limited to the 3 recommended ones). While working on the sets, you should first try to work each problem alone consulting only the lecture notes and the textbooks you already read. If after about 30 minutes you still feel stuck, you may consult other books and people (friends, TAs etc) enough to get you unstuck. But you should write your solution on your own (not while looking at or listening to someone else's solution or solution outline), so that you actually think through it. After you have what you think is a solution, you may check your final answer in any way you see fit. If you find it is wrong, however, you should rework the problem as described above. Slavishly copying solutions from other people, books, previous years' solution sets, etc is
1. pointless (you came here to learn how to do things, right? Copying letters and shapes you already mastered in kindergarten) and
2. an honor code violation in this course.
Problems for which computer assistence of a numerical or symbolic sort is allowed (and sometimes even required) will be explicitly indicated.

Extensions only in exceptional circumstances: see Andy Monro
Hand in what you can by the due date for full credit.
Work (including portions of a set the rest of which was handed in on time) accepted without extension for 50% credit up to one week late. No credit after that.
Solutions available online immediately after the due date.
Please report suspected errors in problems or solutions to Dave Goulet.

### Exams

Midterm and Final available online at 3:30pm.
Completed exams must be signed in to Firestone 307 by 3pm on the due date.
Exams should be completed in an single sitting of 4 hours.
Calculators are not permitted.
Closed-book: with the exception of official lecture handouts and problem set questions, only material written in your own hand (e.g. your homework solutions and class notes) may be used  during the midterm exam.  During the final exam you may also use the offical homework solution sets of 2004 ACM95b/100b [ie the ones posted on this website].
Extensions only with permission of the Dean
A review session will be offered before each exam.
The midterm review session will be 8pm-10pm Weds Feb 4, in 201 E Bridge.
The final review session will be 8pm-10pm Weds Mar 10, in Baxter Lecture Hall (same place the
midtem session moved to).
Here is the instructor's list of what he considers examinable on the midterm.
Here is Dave Goulet's unofficial expanded version of this midterm topic list with examples.
Here is the instructor's list of what he considers examinable on the final exam.
Here is Dave Goulet's unofficial expanded version of this final exam topic list with examples.

The higher of two schemes:
1. 25% problem sets, 37.5% midterm, 37.5% final
2. 50% midterm, 50% final
Please check your graded exams promptly. In the event you are unhappy with grading of your exam, note a new policy: there will be a strict Statute of Limitations on revisions to exam grades:
Midterm grades will not be changed more than one week after the midterms are returned (i.e. after  6:00pm Feb 25).
The graders for the midterm exam are:
Problem 1: Ye Li
Problem 2: Li Ni
Problem 3: Roger Donaldson
Problem 4: David Hoch

Here is the histogram of midterm letter grades and the corresponding numerical cutoffs of midterm scores (best of midterm exam or 1/4 homework+3/4 midterm exam). The cutoffs are nonuniform because they were adjusted to try to fit natural breaks in the distribution of scores (so numerical scores differing by 0.001 would not have different letter grades).

The graders for the final exam are:
Problem 1: Jian Lu
Problem 2: Francesco Ciucci
Problem 3: Gaby Stredie
Problem 4: George Ouyang
Problem 5: Jing Xiong
Final exam grades will not be changed after 6:00PM April 5 (i.e. one week after the end of vacation).

Here is a histogram of final exam scores, and the histogram of final ACM95b letter grades and the corresponding numerical cutoffs (best of  exams average or 1/4 homeworks avg+3/4 exams avg),  excluding I's and other special cases. The cutoffs are nonuniform because they were adjusted to try to fit natural breaks in the distribution of scores (so numerical scores differing by 0.001 would not have different letter grades).

### Ombudspersons

The following students in the class have kindly volunteered to serve as ombudspersons for the course:

Charles McBrearty (So, Ricketts)  <mcbreart@its.caltech.edu>,
Cindy I-Jung Chen (Jr, Ruddock)  <ijung@its.caltech.edu>,
Haluna Penny Gunterman (Jr, Lloyd)  <haluna@caltech.edu>,
Lyle Joseph Chamberlain (So, Page) <lyle@its.caltech.edu>,
Marcus Q. T. Ng (So, Blacker)  <marcusn@its.caltech.edu>,

Please give them your feedback (e.g. on which homework problems are too tedious, how they could be modified, which ones are good and should be emulated, which ones took an unreasonable amount of time, which parts of lectures were good and which were dull or confusing, etc). They will collate and transmit your feedback to the instructor and TAs.

### Problem Set and Exam Schedule

 Available (link is problem set) Due (link is solution set) PS1 Friday Jan 9 Friday Jan 16 PS2 Friday Jan 16 Friday Jan 23(5d corrected 1/30/04) PS3 Friday Jan 23 Friday Jan 30 PS4 Friday Jan 30 Friday Feb 6 Midterm Friday Feb 6 Tuesday  Feb 10 PS 5 Friday Feb 13 Maple example (.mws) Maple example (.html) Mathematica example (.nb) Mathematica example (.html) Friday Feb 20 (minor revision Feb 26) PS6 Friday Feb 20 (hints added Feb 24) Friday Feb 27 (minor revision Mar 4) PS 7 Friday Feb 27 Friday Mar 5 PS 8 Friday Mar 5 (hint added to 3e and numerical errors in 4 e,f fixed Mar 10) Weds Mar 10 (full credit till Fri Mar 12) Solution to Probs 1-3. Solution to Prob 4. Final Friday Mar 12 Weds Mar 17

### Recitation (Tutorial) Sections

Attend any section of your choosing.  Recent changes are highlighted.