Honors Mathematics II

Project

If you choose a term project, you will explore an area of mathematics beyond what will be covered in the course. You may choose the topic from the suggested topics listed below based upon your interests, but your project must include significant mathematical content that illustrates your understanding of the relevant material. Suitable content might be several worked examples, an explanation of a proof, or detailed description of a mathematical system. This should not be directly copied from source material.

The term project should be a substantial piece of work, which you will be proud of, and you should be able to consider yourself at least an expert layman on the subject once you are done. While there is no set word limit or page count, 3000 words should be considered a minumum for the final version of the project.

A web-site prepared by my wife (Kimberly Chapman) containing information, guidelines and helpful hints about how to go about a major project like this can be found at http://www.kimberlychapman.com/essay/essay.html. Everyone doing the project is expected to have read those pages.

Suggested Topics

The following are some suggested topics for projects:

The Theory of Limits

Although calculus was developed by Newton and Leibniz using infinitesimal techniques similar to the ones used in class, calculus these days is usually done using limits. Discuss the history of the development of limits as a reaction to criticisms of infinitesimal techniques. Define limits in solid mathematical terms, and explain (with examples) how they work, and how the derivative and integral can be defined using them.

You should go into much more depth than the text does in its coverage of limits

Fractals

Many natural phenomena are described by functions which cannot be differentiated. Common examples of these are shapes and curves called "fractals." Explain what a fractal is, discuss their development, and why fractal curves cannot be differentiated. Use many examples, and discuss how the theory of fractals has been applied to real-world problems.

Probability and Statistics

Integration and differentiation play a key role in the theory of probability and statistics, particularly when considering continuously varying quantities. Discuss, with examples, how calculus is used in this area of mathematics. Explain why it is a problem that the function e^x2 cannot be integrated as an elementary function.

Writing Style

As honours students your written expression should be of a very high standard and reflect your maturity. While content is important, writing style will be given significant weight in the assesment of all parts of the project. You should ensure that your presentation is clear and concise, your spelling and grammar are correct, and the project clearly divided into thematically related sections.

Suggestions on citation and bibliographies are included at the end of this document.

Plagiarism

You should be especially wary of plagiarism. Plagiarism is attempting to pass off someone else's work or ideas as your own. At one extreme is wholesale copying of anothers' work, which hopefully no one in this class would dream of doing, but it also includes failure to properly cite your sources, or lifting quotes from a source without properly indicating them. Plagiarism is tantamount to stealing another's work and as such is a serious form of dishonesty. Penalties for plagiarism range from taking marks off to expulsion from university, depending on the severity of the case.

Due Dates

The term project has three parts to it:

Preliminary Report

A preliminary report is due on Friday, February 29th. This should be a 2 or 3 page summary or outline of what you will cover in your project, together with a bibliography of all the references that you will use. This will be worth 15% of your course grade. If the preliminary report is unsatisfactory, you will have to take all the examinations for your grade.

The preliminary report will be reviewed when the final draft is handed in, and the actual material covered compared with what the preiminary report said would be covered. It will count against you if you say that you will cover something in your preliminary report and then do not in the final draft.

Complete Draft

A complete draft of your project, together with the graded copy of the preliminary report, will be due Wednesday, April 19th. This is worth 15% of your total grade. If you do not hand this in, or the draft is unsatisfactory, then you must take the remaining two examinations.

Final Draft

The final draft, together with the graded copies of the preliminary report and the complete draft, will be handed in Friday, May 9th (the last day of classes) and will be worth 30% of your grade. The preliminary report and complete draft will be compared with the final version. Suggestions from the grading of the complete draft which are ignored without reason will count against the final grade.

All deadlines are hard. If you miss a deadline without a valid reason, you will have to take the examinations instead.

Bibliography and Citation

You should make sure you document all your sources in an annotated bibliography, and cite them within the paper whenever you refer to them. Note that you must cite even when you are paraphrasing a source, not just when directly quoting. Your bibliography should give enough information for a reader to find the referenced work, as well as a couple of sentences describing the work and its usefulness in your research.

While the World Wide Web is a rich source of information, web pages are not sufficently reliable to be the sole source for scholarly work. You must find as many relevant book or article references as you can as part of your research. If you do use a web site as a source, be sure to indicate clearly which web pages you are referring to when citing.

I do not care how you indicate citation or organise your bibliography, as long as it is consistent, clear and gives all required information. Different disciplines have different styles, but if you use a standard style you should be fine. Examples of sufficient citation would be:

Jones (53) asserted that the widgetiscope contributed to the deaths of a number of widget watchers by emitting excess levels of baryonic radiation. More recent investigation by James T. Bruce of baryonic radiation emmissions from Smith's original widgetiscope show that "emissions of tau widgetrons were below background levels, while mu and zeta widgetrons were not detected at all." (Bruce and Bruce, 75) The deaths have not been adequately explained, but feral widget attacks are now considered to be the most likely cause. The controversy surrounding Jones' allegations may have been a contributing factor in the depression that led to Smith's suicide. (Shelia, "deaths.html")

This means that Jones' work talks about the deaths on page 53; the direct quotation is taken from page 75 of Bruce and Bruce; and the last part of the example refers to the page "deaths.html" on Sheila's web site.

These references might be listed in the bibliography as:

Bruce, James T. and Bruce, P.D.Q., "Baryonic Radiation Emissions from Widgetiscopes," Journal of Widgetronic Science, 19 (1), 12-102, 1996.

This paper reports experiments performed to detect baryonic radiation levels caused by Smith's original widgetiscope. Bruce and Bruce's results were important in refuting claims that the widgetiscope was responsible for a number of deaths.

Jones, I., Widget Watching in the Temple of Doom, Knopf, 1937.

Jones' memoir recounts his days as a widget watcher in pre-war India. The book describes the politics of New York Widget Watchers' Society, who financed Jones' expedition, and their reaction to Smith's invention. The possibility of the widgetiscope being the cause of a number of deaths is also raised.

Sheila, E., "The Life and Times of Joe Smith," http://www.widgets.org/smith/

This site gives a detailed biography of Joe Smith. It also includes an archive of Smith-related material, such as scans of the blueprints for the original widgetiscope.

Bibliographies are traditionally listed in alphabetical order by Author, and then from the oldest work to the newest work or alphabetically by title.

The first entry is a jornal article which appeared on pages 12-102 of volume 19, issue 1, of the Journal of Widgetronic Science, which was published in 1996. The second entry is a book published by Knopf in 1937 (some reference styles will include the city of publication as well). The last entry is a web site. If a web site includes multiple pages the bibliography should list the basic URL and if the web site is cited as a source the citation should indicate which page on the web site is being referred to.