Applications of Exterior Differential Forms 
Syllabus and Commentary

General InformationThis course will be taught by inclass lecture as well as via a computer connection from your dorm or your home to the World Wide Web. Much of the course will involve mathematics that can be symbolically manipulated via Maple (or Mathematica). Equations developed in the Lectures (or lecture notes) will be programmed in Maple syntax, such that the student can download the programs to his local computer, execute the programs with visual graphic output, and then have the ability to modify the equations and parameters to study new situations that might interest him. SW20 from TCI Software is used to generate the lecture notes and articles with fancy equations. This scientific word processor is amazing, for it is linked with the Maple program. You write an equation and then ask Maple to solve it  and it does, putting symbolic solutions right into your article.
You will need to download a FREE copy of TCI Scientific Notebook to be able to
read and print the .tex files that contain the homework. It is much better and
faster than the Adobe Acrobat reader (which is also free). You will need to
download a FREE copy of the Adobe Acrobat reader to be able to read (and print)
.pdf files. Try out the formats by clicking on
Cartan's Repere Mobile
tex file 16k
Cartan's Repere Mobile
pdf file 64k
You can get the FREE readers and other software by clicking on the icons below
The U of H has a site license for Mathematica, so it is available on
almost every PC and MAC at the University. Negotiations are under way to get a
site license for Maple, which I think is a superior product, especially
now that it has been integrated with the terrific Scientific Latex Word
Processor program, SWP30.
The University of Houton offers a free INTERNET dial up account to all students
and faculty. Contact University Services at 7431411 and get on line now with a
PPP account.
PrerequisitesIt would help if you know a bit about vector analysis and understand what is meant by grad, div and curl. By the time an Engineer or Physical Scientist is in his third year at the university, he has been exposed to a lot of the standard vector methods. You should also understand the concepts of partial differentiation, know a little about linear algebra and matrices, and have had some exposure to differential equations. The prerequisite math level of the course will be that of a third year course in advanced calculus.
Course ContentThe purpose of this course is to stimulate interest in topological aspects of science and engineering. The course will give an introduction to vector and tensor analysis, differential geometry, differential topology, and how these mathematical methods can be utilized in engineering and physics.
In particular Cartan's methods of Exterior Differential Forms will be applied to
problems in hydrodynamics, electromagnetism, thermodynamics, and relativity. The
course will rely heavily on computer techniques for solving problems, and
generating graphic displays of the concepts of differential geometry and
differential topology.
The online lecture notes will have philosophical introductions to the various
topics (on the net) and will be linked to more extensive and technical pdf
files. Links to useful Maple worksheets will be distributed with the notes. A
list of the Maple worksheets is also maintained, and hopefully will have
contributions from users of this course.
Text Book and ReferencesThe required text book for is A Course in Mathematics for Students of Physics Vol 1 and Vol 2., P. Bamberg and S. Sternberg. Cambridge University Press (1992)
It is suggested that you buy also:
Differential Forms by H. Flanders
Differential Geometry by Dirk Struik.
Schaum's Outline General Topology by S. Lipschutz These books are now in (relatively) inexpensive Dover paper back form.
The U of H has a site license for Mathematica, so it is available on almost
every PC and MAC at the University. Negotiations are under way to get a site
license for Maple, which I think is a superior product, especially
now that it has been integrated with the terrific Scientific Latex Word
Processor program, SWP30.
