Math 46: Introduction to Applied Mathematics - SPRING 2009

Alex Barnett. Kemeny room 206, tel 6-3178, email: m46s09 at

shock wave forming in traffic flow (Keyfitz)

Eigenfunctions in a planar drum (Barnett/Betcke)

Mathematics is being increasingly applied throughout the physical, biological and social sciences ...the skill to do this successfully is highly sought after! This course provides essential analytical ("pencil-paper-brain") tools for mathematical modeling of the real world, and solving (or approximately solving) the resulting equations.

We start with dimensional analysis and scaling, which allows you to say a surprising amount without solving any equations at all! Most of the time the equations arising in nature are differential equations. With ODEs we explore perturbation methods, that is, asking how a solution changes if the equation changes slightly from a known case; this allows scientists to understand approximate solutions before they sit down at a computer. The "opposite" of an ODE is an integral equation, which introduces us to operators, key objects in any math toolbox (melding calculus and linear algebra). An application will be de-blurring of (1D) images. For PDEs we will cover examples both linear (such as heat flow and waves, building on Math 23) and non-linear (such as reaction-diffusion equations for bacterial growth). To analyse linear PDE we learn tools such as Greens functions and the Fourier transform. We might get to water waves (linear dispersive PDE) or shock waves in traffic flow (a nonlinear PDE). There's also a little numerical plotting (eg Matlab or whatever you prefer), but numerical methods are not a focus of the course.

Lectures / OH: Kemeny 108, MWF 1:45-2:50pm (period 2), important to attend since we'll do lots of worksheets together. I strongly recommend you read the material in the book in advance of the lecture. X-hour is 1-1:50pm Thurs, and I will use intermittently for: Matlab sessions, review, problem-solving sessions, catch-up lectures, etc. Do not schedule anything regular in this X-hr; I expect to use about half of them. I encourage you to come to office hours: Mon 3-4pm, Tue 1-2pm, and Fri 11am-12.

Required book: Applied Mathematics, Third Edition by J. David Logan (Published by Wiley Interscience, New York, 2006). This otherwise excellent book has a moderate list errata which I suggest you copy in to save later confusion. Available at Wheelock Books, etc.

Homework: 9 weekly HW's due Wednesday at start of lecture. I strongly suggest you collaborate, and try at least some of the relevant homework problems before the next lecture (leaving it all for Tuesday night risks you getting left behind in this fast-paced course.) Please make your working/reasoning as clear as you can, write clearly, don't be scared of using lots of space on the page, and staple your work. Late homework will not be accepted (unless by prior arrangement for a valid, and exceptional, reason). Your lowest HW score will be dropped, ie you get a `free' HW.

Exams: I will try to give you ample time to complete exam questions. However, the only key is to practise, practise, practise. (Also read this).

Honor principle. Exams: no help given or received. Homework: group discussion and collaboration on problem techniques is great and helpful. Write-ups must be done individually (ie no copying).

Grades: Will be based on HW 30%, Midterms 2*20%, Final 30%. Note the HW is quite heavily weighted, and is the main chance you get to practise and get feedback, so stay on top of it. Grades in Math 46 are not curved; other students' good performance will not hurt your grade. (So please work together and help each other out!)

Special needs: I encourage students with disabilities, including "invisible" disabilities like chronic diseases and learning disabilities, to discuss with us any appropriate accommodations that might be helpful. Let me know asap, certainly in first 2 weeks. Also stop by the Academic Skills Center in 301 Collis to register for support services.

Private tutoring: Tutor Clearinghouse may have private one-on-one tutors available for Math 46 (although this is rare in advanced courses, so come and talk to me if you feel lost and no tutors are available). If a student receives financial aid, the College will pay for three hours of tutoring per week. If you would like to have a tutor, please go to 301 Collis and apply as early as possible.

Religious observance: Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me before the end of the second week of the term to discuss appropriate accommodations.