APAM 1601 Site Information

Introduction to Computational Mathematics and Physics

Prof. Michael Mauel
Email: mauel@columbia.edu


Welcome to the APAM 1601 class information site.

Introduction to computational methods in applied mathematics and physics: Students develop solutions in a small number of subject areas to acquire a first taste in the practical use of computers in solving mathematics and physics problems.

APAM1601 does not require prior programming experience (but prior computer experience and talent are helpful.) Topics change from year-to-year, and only a limited number of topics (typically four per term) are selected for discussion and investigation. Topics range from classical and modern physics and applied mathematics, but the course is not meant to cover these areas broadly. Instead, each topic will be self-contained and limited in scope. We try to make topics interesting and absorbing, and they will amplify and expand on a student's knowledge acquired during your first year of physics and mathematics course work. The goal of this course is to provide some depth in select topics instead of providing a general (but shallow) overview of an entire subject area.

Examples include elementary interpolation of functions, solution of nonlinear algebraic equations, curve-fitting and hypothesis testing, wave propagation, fluid motion, gravitational and celestial mechanics, chaotic dynamics. (APAM1601 is usually taught by a team of two professors, an applied physicist and an applied mathematician. This semester, the course will have a stronger computational physics emphasis because of the interests of the instructor.) The basic requirement for this course is one year of college level Calculus and Physics; programming experience is desirable but not required. Students will gain confidence preparing, understanding, and presenting computer-assisted solutions to elementary but relevant problems.

This course uses Mathematica combined with detailed classnotes.

Why Mathematica? Mathematica is a very powerful system for evaluating expressions and graphing results. It's disadvantage is that it is commercial software, but it's advantage is that it is available for all major operating systems and available widely at Columbia University. The most important reason I selected Mathematica is that it is an interpretive system: Mathematica instantly responds to every expression you write. Additionally, Mathematica notebooks contain a perfect record of your interaction with the computer. Save your notebooks, edit them, and re-execute them. In the beginning, the syntax of Mathematica will seem complicated. Later, it will become second nature. Experience using Mathematica will be developed as the complexity of the subject material is advanced.

Mathematica is available on all of Columbia's computer labs. You can also purchase a license for Mathematica to run on your personal computer for as little as $44.95.

(If you are interested in submitting your projects in Matlab or any other programming language or system, you are free to do so. However, class time is limited, and I will not be discussing these other systems in the course.)

Note: Our new class time is Tuesday and Thursday from 11:40 AM through 12:55 PM in Room 233 of S. W. Mudd.


This Web Site is a basic resource for APAM1601.

An introduction to the course is available in Adobe PDF format.

Each link below is to a Mathematica notebook (with ".nb" filename extension). You must download the link file and be certain that your browser does not append a ".txt" filename extension. For example, if you download the file "1-Basics.nb" to your local computer, then you can directly read the file using Mathematica and begin running, exploring, and modifying your programs.

Lecture Date Notebooks

January 22

January 24

How to use Mathematica?

January 29

January 31

February 5

February 7

Review introductory notebooks: 2-Pendulum.nb and 1-Numbers.nb

Harvard Professor emeritus Owen Gingerich article in Physics Today (2011) entitled "The great Martian catastrophe and how Kepler fixed it"

Planetary Motion, Nonlinear Ordinary Differential Equations

February 12

February 14

First Quarter Research Project

February 19

February 21

February 26

February 28

Working with Random Numbers and Statistics of Physical Behavior

Fractals in the News…

March 5

March 7

March 12

March 14

March 18-22 Spring Break
Second Quarter Research Project

March 26

March 28


Digital Signal Processing

Digital Fourier Transform (DFT), signal processing, and information compression:

Wikipedia's excellent article about information theory is here. Please read the biography of Claude Shannon. When Shannon was 21 years old (1937), he wrote a masters thesis showing that any logical numerical relationship could be constructed with Boolean logic. More plainly, he showed that digital computing works. Regarding our research topical area, Shannon developed the theory defining the relationship between information entropy and data compression.

Sound and Graphics Files
Third Quarter Research Project

April 3

April 5


TIFF Image Files

April 2

April 4

Quantum Mechanics and Partial Differential Equations

April 9

April 11

Fourth Quarter Research Project
  • Due: Tuesday, May 7
  • UnderstandingQuantum.nb
  • UnderstandingQuantum_Demo.nb (This is a short example of the 1D quantum scattering problem. You'll get different results for nGrid = 100 or for nGrid = 200. Note the clever algorithm in findMaxTransmission[e]. It changes the number of time-steps of the Crank-Nicholson integrator according to the speed of the particle.)

April 16

April 18

April 23

April 25

  • Review and problem solving

TA and Help

Christopher Choi. is the graduate student TA for this course. Christopher studiessolid-state physics and he is very knowledgeable about computing and mathematical modeling. Send Christopher emails (at <csc2174@columbia.edu>)if you want some help and advice about your exercises, project reports, or programming hints.

Mathematica Links

The Wolfram site is a source of considerable information on Mathematica. Columbia University has a license agreement with Wolfram for Mathematica.

Prof. Nicholas Giordano's course website contains information and source listings from the suggested text book, Computational Physics.

Schedules for the AcIS computer labs are found here.

Professor Michael E. Mauel
Department of Applied Physics
Columbia University

Go to Prof. Mauel's HomePage