You can submit your work either as a traditional word-processed document, or as an .html web page containing about 5 pages worth of text if you like. I will be looking primarily for content in either case -- see comments below. If you go the .html route, while images and hyperlinks to other WWW pages are great to include, they alone will not be sufficient.
Papers may be submitted at any time up to the Friday of the week before the day of the final exam for the course: Friday, May 1, 1998.
In working on this paper, you should follow the same procedures you would follow in preparing a research paper for any other course. Your grade will depend on the thoroughness of your research, the degree of independent thought about the subject revealed through your work, the organization of the paper, and the quality of your writing. Do not assume that because this is a mathematics paper writing does not matter! A word to the wise: I tend to be rather severe when I read papers with large numbers of misspellings, or many grammatical and other technical errors. The same thing happens with papers consisting largely of masses of undigested quotations from your sources, particularly when those quotations involve technical discussions of mathematics you do not understand fully. To avoid having this happen to you, take me up on the following offer: I will be happy to read a preliminary draft of your paper and give you comments. The Writing Workshop is also a good resource for help on organization and other more mechanical writing issues.
In the suggested topics below, all citations refer to the list of references at the end of this handout. These books are all on reserve in the Science Library in Swords Hall for this class. There are many other histories of mathematics in the library which may be useful to you as well. Mr. Stankus, the science librarian, will be more than happy to assist you in finding other books on the history of mathematics if you need help.
At the end of the paper, include a full list of all the references you consulted. These references may come entirely or in part from the reserve list below; also include all sites on the Web you use to do your information gathering. For any Web sites you consult, please give a page title if you can identify one, and the complete URL in a reference like this: http://www.somewhere.edu/somepage.html.) Include foot- (or end-) notes identifying any direct quotations you use, or any statements of opinions which are not your own. Failure to follow these guidelines constitutes plagiarism, a serious academic offense, which will result in an automatic grade of F on this paper. If you have questions about the meaning of this, please consult with me. I will be glad to try to answer any questions you may have on these topics, and help you in the preparation of these papers.
The qualitative approach we took in this class to critical points and flow lines for vector fields in the plane is largely the creation of the French mathematician Henri Poincar'e, who lived from 1854 to 1912. Poincar'e introduced this approach to study the properties of solutions of an ordinary differential equation dy/dx = Q(x,y)/P(x,y) but the problem of finding solutions of such an equation is essentially equivalent to the problem finding flow lines of the vector field F(x,y) = (P(x,y),Q(x,y)), since if alpha(t) = (x(t),y(t)) is a flowline, then on that curve dy/dx = (dy/dt)/(dx/dt) = Q(x(t),y(t))/P(x(t),y(t)) = Q(x,y)/P(x,y). Hence, provided that y can be defined implicitly as a function of x along the flow-line, the flowline becomes the graph of a solution of the differential equation. For this paper, you would research Poincar'e's life and contributions to mathematics, with special attention to his qualitative work on differential equations. Note: Poincar'e was an extraordinarily prolific mathematician, and he actually studied many different topics even within the subject of differential equations. Some of them are quite a bit more advanced than others, and it would be best not to try to describe some of them in technical detail since the mathematics involved is more advanced than anything you have seen. Instead, address the following questions as you are writing your paper. First, what physical problems was Poincar'e trying to study and why were the notions of stability and instability important for him? What does the Poincar'e-Bendixson Theorem say? What "hot topic" of the last ten years is largely prefigured in Poincar\'e's work? The biography of Poincar'e in the Dictionary of Scientific Biography is a good place to start. (You should also be able to find quite a bit of information on Poincar'e at various sites on the Web.) Bell's book has a chapter on Poincar'e, Kline's book has a chapter on Differential Equations 1800-1900 with quite a bit of relevant material. Stewart should help you with the last question above.
Much of the algebra of vectors in 2- and 3-space that we have discussed in this course was first developed in the early to mid-19th century. For this paper, you would research the contributions of two mathematicians, W.R.Hamilton, H.Grassmann, and the physicist J.C.Maxwell. Among the mathematical topics you should discuss are: How did people think about algebra at the start of the 19th century? Why were people interested in the complex numbers and in ``hypercomplex number systems'' and what did they hope to do with them? How did Hamilton discover the algebraic system that he called the quaternions? Describe the quaternion addition and multiplication operations, giving examples. What are some of the unexpected properties of these operations? (That is, how do they differ from the ordinary arithmetic operations on rational numbers, or the real numbers?) What was Grassmann's point of view? What have we taken from him in this course? How did Maxwell use the new vector algebra? A good reference for this topic is Chapter 32 of Kline's book. Bell, and/or various sites on the Web would be good for biographical details on Hamilton and Maxwell especially.
In the analysis of data obtained from experiments in the physical or social sciences, one important technique is the method of "curve fitting". For example, suppose that it is suspected that some variable y depends linearly on another variable x (that is, that there is a relation y = mx + b for some m and b). From several observed data points (xi,yi) we could try to estimate the values of m and b leading to a line that best fits the experimental data. (More complicated polynomial or exponential/logarithmic relations between y and x can also be studied by the same methods.) For this topic, you would learn the mathematics behind the least squares method, the most commonly used technique to find the coefficients of the linear (or polynomial) function whose graph comes closest to passing through the data points. (In social sciences literature, this process is often called linear or polynomial regression.) First, show how the problem can be rephrased in algebraic terms. Discuss the geometric meaning of the vector of coefficients for the best fit line (or polynomial), using the language of lengths of vectors. Show how to compute the best fit line (or polynomial), and illustrate your statements with some examples (the more realistic, the better!) For example, if you have used this technique in a lab write-up in another course, you could present your experimental data, together with the results of the least-squares computation, and discuss the interpretation of the computed coefficients m and b in that experiment. See me for suggestions for references on this topic. The major mathematical tools you will need are: 1. partial derivatives, and 2. some elementary facts about solving simultaneous systems of linear equations.
For parametric curves alpha(t) in the plane or in space, there is a natural notion of curvature. If T = alpha'(t)/||alpha'(t)|| is the unit vector in the direction of the tangent vector, then we can measure curvature by looking at the rate of change of T as a function of arclength measured along the curve. At the end of the 1700's several geometers, including Monge, Euler, and Meusnier, introduced the idea of studying the curvature of a two-dimensional surface at a given point by considering curvatures of the "normal sections" of the surface--the intersections of the surface with planes containing the normal vector at a given point. Among the discoveries made at this time were the surprising facts that the curvatures of the normal sections all lie between a maximum k1 and a minimum k2, and in the case where k1 and k2 are different, the planes which cut out the sections with curvatures k1 and k2 meet at right angles. These statements are known today as the theorem of Euler and Meusnier. For this paper, you would research the history and mathematics of this work. See me before starting if you choose this topic; I will be able to point you in the direction of good sources for this material.