What will I study in this module?

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 Élie 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.

How will the module be assessed?

  • The end of semester exam is worth 55% of the total module assessment.

    A sample exam paper can be found here.
  • The continuous assessment contributes 45% to the total module score. It consists of three in-class tests (each consisting of five problems based on problems from the problem sheet) and three online Okuson homeworks. Each of the three Okuson homeworks contributes 5% to the continuous assessment component, and each of the three in-class tests contributes 10%.

Live lectures, on-campus tutorials, textbook, and contact

  • Lecturer: Prof Graham Ellis
  • Lectures: Mon 11am on Zoom and Wed 11am on Zoom. These will be recorded and made available for repeat viewing via Youtube links that will be posted on this page.
  • Tutorials: The class will be split into three groups A, B, C. You'll find your group on Blackboard. Each group is invited to attend a weekly on-campus tutorial at the following times and venues, starting on Tuesday 29 September:

    Group A: 10am Tuesday, O'Flaherty Theater

    Group B: 2pm Tuesday, O'Flaherty Theater

    Group C: 11am Friday, O'Flaherty Theater

    To comply with Covid room capacity and contact tracing requirements, students are only allowed to attend the tutorial for their own group. Attendance will be taken at tutorials using Qwickly on Blackboard; the attendance record will only be used for Covid contact tracing; it will not be used in any part of the module assessment.
  • 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.
  • 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.

Homeworks and deadlines

Problem sheet: available here.
Online homeworks: available here.

Friday 16 October, 5pm First Okuson homework sheet
Wednesday 28 October, 11am First test
Friday 13 November, 5pm Second Okuson homework sheet
Wednesday 25 November, 11am Second test
Friday 04 December, 5pm Thursday Okuson homework sheet
Wednesday 16 December, 10am Third test

Recorded lectures and notes

Lecture Notes
Lecture Summaries
1
Video of the lecture.

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
Video of the lecture.

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
Video of the lecture.

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.
4
Video of lecture.

Introduced the notions of a differential 0-form and differential 1-form in several variables. Focused 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.

Also gave the definition of the derivative of a function f:Rn --->Rm.
5
Video of lecture.

Used the example of a particle in a constant force field to illustrate a (constant) differential 1-form. Then used the marginal costs of acquiring/relinquishing assets from an investment portfolio to illustrate a (constant) differential 1-form. Went on to explain how a differential 1-form can be viewed as an infinite collection of vectors, one vector associated to each point in space. Finished by introducing the integral of a 1-form on a connected, oriented 1-dimensional region as the "work done" in moving a particle from the initial boundary point of the region to the final boundary point of the region.
6
Video of the lecture.

Gave the formal definition of the integral of a 1-form w along an oriented connected 1-dimensional region S. Illustrated the definition by explicitly calculating two integrals.

The definition can be interpreted as work done in moving a particle along an oriented curve S under a force field represented by a 1-form.

The definition can also be interpreted as the total cost of restructuring an investment portfolio, where the restructuring involves a sequence S of sales and acquisitions, and where the marginal cost is described by a 1-form.
7
Video of the lecture.

Computed the work done in moving a particle once around an ellipse, anti-clockwise, in a force field defined by a differential 1-form on 3-dimensional space. In the computation we used: (i) the definition of an integral of a 1-form to simplify the problem to one involving a 1-form in just two variables; (ii) a substitution to reduce the two variables to a first year integral involving just one variable t. Then went on to define the partial derivatives and the total derivative of a 0-form in several variables. So we now understand both sides of Stokes' formula for a 0-form w in several variables. Next lecture we'll give a proof of the formula in this case.
8
Video of the lecture.

Proved Stokes Theorem for differential 0-forms (though the proof requires the Chain Rule for differentiation which will be covered in a subsequent lecture!!). Illustrated how Stokes Theorem can be used to calculate integrals of certain 1-forms.
9
Video of lecture.

Explained the Chain Rule for finding partial derivatves of composite functions.
Then "recalled" that a function f(x,y) is continuous at a point (xo,yo) if f(xo,yo) equals the limit of f(x,y) as (x,y) ---> (xo,yo). This definition works for functions of an arbitrary number of variables, not just two variables. Gave some examples on continuity of functions of several variables. Defined what it means for a function to be "continuously differentiable" in a region. Mentioned that continuous differentiability implies differentiability.
10
Video of the lecture.

Introduced the definition of an integral of a constant 2-form over an oriented planar triangle. Some example integrals will be calcuated next lecture.
11
Video of the lecture.

I started with some examples of calculating integrals of constant 2-forms over oriented planar triangles in 3-dimensional space.

I then started to give the definition of the integral of a (not necessarily constant) 2-form over an oriented surface in 3-dimensional space. I forgot to plug the power into my laptop, and so everything crashed half way through this definition. I'll give the definition in the next lecture, starting from the start and with the power cable plugged in.
12
Video of the lecture.

Defined what we mean by the integral of a differential 2-form over an arbitrary oriented surface S.

To re-inforce our understanding of this definition a couple of integrals were evaluated.

It is worth repeating: differential 2-forms can be regarded as symbols which obey certain algebraic rules (such as dx ∧ dy = -dy ∧ dx) and which can be integrated. We have a full and precise understanding of what the integral of a 2-form represents, and we'll not worry too much about what a 2-form itself represents. The algebraic rules for 2-forms are derived from properties that clearly hold for their integrals
13
Video of the lecture.

Began with some explanation of the second online homework.

Then considered a business maths problem whose solution is obtained by integrating a differential 2-form over a planar region.

Then explained the rules for differentiating a differential 1-form w in order to obtain a 2-form dw.

I'm taking the view that: (i) differential n-forms are things represented by symbols; (ii) I'll not worry too much in lectures about what they; (ii) the online homeworks are devoted to explaining what n-forms are; (iv) I'll be very precise in lectures about what is meant by the integral of an n-form over an oriented n-dimensional region; (v) in order to calculate integrals of n-forms we'll need to know the rules for manipulating and differentiating n-forms; (vi) these manipulation rules are forced on us by our understanding of an integral of an n-form and by the desire to have Stoke's formula hold.

So at the end of the module we'll be able to integrate over n-dimensional regions using Elie Cartan's language of differential forms, a language that is perfect for:
  • providing one simple and unified explanation of a range of what, in other approaches, would appear to be ad hoc notions and results (such as curl, div, grad, the Fundamental Theorem of Calculus, Green's Theorem, Gauss' Theorem, ...)
  • providing a generalization of basic ideas of calculus to n-dimensions,
  • giving a concise description of electromagnetism.

  • 14
    Video of the lecture.

    Provided some justification for the rules for differentiating 1-forms w by showing that they suffice to establish Stokes' formula (for w a 1-form in two variables x,y and S an oriented region in the xy-plane).
    15
    Video of the lecture.

    Showed that under a mild hypothesis on a differential 1-form w we have d(dw)=0. Gave an example which used this equation. Then quickly covered the basics on differential 3-forms, and their integration, and the total derivative of a 2-form. The treatment was such that it equally applies to differential k-forms (though the 1-dimensional notion of "length", and the 2-dimensional notion of "area", and the 3-dimensional notion of "volume" has to be replaced by the k-dimensional notion of the determinant of a k×k matrix).
    16
    Video of the lecture.

    I explained that a differential 0-form w(x,y,z) in 3 variables, in conjunction with some constant k, can be thought of as the surface S defined by
    w(x,y,z) = k.
    I went on to explain that, when the 0-form w is viewed in this way, the derivative dw is just a unit normal vector to the surface S.

    The explanation began with a review of the dot product of vectors. It also included the alternative terminology grad(w), or gradient of w, for derivative dw of the 0-form w.
    17
    Video of the lecture.

    Calculated the tangent plane to a surface. Then introduced the curl of a vector field -- it is just another name for the total derivative of a 1-form. Gave an interpretation of the curl (without proof).
    18
    Video of the lecture.

    Defined and interpreted the divergence of a vector field on 3-dimensional space.

    Then stated Green's Theorem in the plane, Stokes' Theorem (a generalization of Green's Theorem), and the Divergence Theorem. They are each just special instances of Stoke's formula!
    19
    Video of the lecture

    Verified Green's Theorem in the plane on a particular example. Then derived a general formula for the area bounded by a simple closed curve in the plane, and used it to calculate the area of an ellipse.
    20
    I forgot to press "record video". But no material from this lecture will appear on the exam.

    Described electromagnetism by introducing a 1-form H, a 2-form D and a 2-form E^dt+B and a 3-form J satisfying three equations.

    If you are a financial maths student wondering how on earth the theory of electromagnetism could ever be of relevance to you, then take a look at the Wikipedia page on econophysics.
    21
    Three lecture slots were taken up with class tests, and one slot was a bank holiday.
    22

    23

    24

    Supplementary material

    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!

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