Module Content

The module explains Stokes' formula

∂S ω = ∫S ∂ω
where:
ω is a differential p-form in n-variables;
S is a "nice" (n+1)-dimensional region in Euclidean space ℝn+1;
∂S denotes the boundary of the region S;
∂ω denotes the exterior derivative of the form ω

It's a good topic for a university in the West of Ireland since George Gabriel Stokes was born and brought up in Skreen, County Sligo. The formula is best explained using the differential forms language of lie Cartan. The module covers
  • the case p=0 (8 lectures),
  • the case p=1 (8 lectures),
  • the case p≥ 2 (8 lectures),
  • and explains how Stokes' formula can be applied to

  • the usual problems of multivariate calculus,
  • financial mathematics,
  • electromagnetism.

Module Coordinates

  • Lecturer: Prof Graham Ellis
  • Lectures: Mon 11am in Anderson and Wed 11am in AM200
  • Tutorials: 6pm in ADB-1020 on Tuesdays and 12pm on Fridays in BLE 1006 (Block E - old Engineering Building). The tutor on Tuesday is Kelvin Kileen and the tutor on Friday is Adib Makroon.
  • Recomended text: Most of the homework questions are taken from Murray Spiegel's elementary level book: Advanced Calculus (Schaum's Outline Series). I recommend that you buy this book.
  • Problem sheet: available here.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA2286 Differential Forms pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

  • The end of semester exam is worth 70% of the total module assessment. It will consist of five questions each with three equally weighted parts. Each question part is worth 9 points and the exam score is capped at 100 points.

    A sample exam paper can be found here.
  • The continuous assessment contributes 30% to the total module score. It consists of three in-class tests, each consisting of five problems based on problems from the problem sheet.

Supplementary Material and News

CLICKER OPINION POLLING may be used in some lectures.

FINANCIAL MATHEMATICS

Are you an FM student asking why we are making you study differential forms? For one answer take a look at this paper on "Applying Exterior Differential Calculus to Economics: a Presentation and Some New Results", and this paper on "Exterior Calculus: Economic Profit Dynamics", or just Google "exterior caclulus" and "Economics" and browse through the many articles and books that you hit. The MA2286 module aims to give you the ability to benefit from such research articles and books in financial mathematics!

STUDENT FEEDBACK

I'll place student feedback here.

Lecture Notes

Lecture Notes
Lecture Summaries
1
I explained that the aim of the module is to understand and apply Stokes' formula using the language of differential p-forms in n variables. Stokes hails from just up the road in Sligo so it seems appropriate to devote a 24-lecture module to his formula. We'll use the language of differential forms because of its elegance and simplicity. In this first lecture I focussed on n=1 variable and p=0 and gave the definition of a differential 0-form in one variable. Differential forms are defined with respect to some nice oriented region S in ℝn. I explained what I mean by such a region for n=1: namely the union of a collection of disjoint oriented closed intervals. I explained what is meant by the (oriented) boundary of such a region and ended with the definition of the integral of a 0-form in 1 variable over the boundary of a 1-dimensional oriented region in ℝ.

2
I explained what is meant by a differential 1-form in one variable. I also explained that the integral of such a 1-form over an oriented region is precisely what we met in the 1st year integral calculus course: it's just an integral of a function of one variable. I ended with a first application of the language of differential forms to a financial maths problem: differential 1-forms were used to model the rate of expendidture and the rate of income in a certain fund raising project. The integral of a differential 1-form gave us the total net income generated by the fund raising project.
3
Explained what is meant by the "total derivative" (or simply "derivative") of a differential 0-form in one variable. So now we understand the meaning of all terms in Stokes' formula for the case n=1, p=0. In this case the formula is just a re-statement of the Fundamental Theorem of Calculus -- a theorem we met in first year. For completeness we recalled the definition of an integral of a function of one variable and then proved the Fundamental Theorem of Calculus. The lecture ended with a 1st year problem about an indefinite integral which was phrased in the language of differential forms. This problem will be finished next lecture.
4
Began by finishing off a first year calculus problem. Introduced the notions of a differential 0-form and differential 1-form in several variables. Focussed mainly on n=2 variables; included an informal discussion on the notion of differentiability and the notion of surface. Motivating examples of 1-forms will be given next lecture.
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