Differential Forms MA2286

Module homepage for 2017-18


Most of the homework questions will be taken from Murray Spiegel's elementary level book: Advanced Calculus (Schaum's Outline Series). I recommend that you buy this book.
However, my presentation of material in lectures is very much based on Harold M Edwards advanced level book: Advanced Calculus: A differential forms approach (Birkhauser). I recommend that you take a look at this book in the library but you don't need to buy it.

Another book that you should certainly take a look at -- its a classic text -- is Michael Spivak's Calculus on Manifolds.

Assessment: The end of semester exam will count for 70% of the assessment. The continuous assessment will count for 30% of the assessment.

Continuous Assessment: The continuous assessment will consist of three equally weighted in-class tests based on the homework/tutorial problems. Each test will consist of questions taken verbatim from the homework/tutorial sheet. Questions on the end of semester exam will also be fairly closely based on the homework/tutorial sheet. Here is a version of the end of semester exam with question details omitted.

Homework sheet is available here. Throughout the semester I'll add extra questions to this sheet.

Lectures take place at 11am Monday in the Anderson Theater and 11am Wednesday in AM200. The lecturer is Graham Ellis.

Tutorials take place at 6pm in ADB-1020 on Tuesdays and at 12pm on Fridays in IT202. The tutor is Adib Makroon.

Student feedback on this module will be posted here.

Previous years: The material covered in the module is similar to previous years. However, the presentation of the material will be a bit different.

FAQ: I came to university to study financial maths and economics, so why are you making me study differential forms?

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 claclulus" 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!


Lecture 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.

Lecture 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.

Lecture 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.

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.

Lecture 5
Use the example of a particle in a constant force field to illustrate a (constant) differential 1-form. Then use 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.

Lecture 6
Gave the formal definition of the integral of a 1-form w along an oriented connected 1-dimensional region S. Motivated the definition by an example involving the total cost of restructuring an investment portfolio where the marginal cost was described by a 1-form. Illustrated the definition by explicitly calculating two integrals.

Lecture 7

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.

Lecture 8
Discussed the Fundamental Theorem of Calculus, aka Stokes' formula for 0-forms w. Used this result to compute some "line integrals". Ended with a short discussion on continuity. A function in continuous is a small change in input results only in a small change in output. Epsilons and deltas can be used to make this precise.

Lecture 9
A function f(x,y) is continuous at a point (x0,y0) if f(x0,y0) equals the limit of f(x,y) as (x,y) ---> (x0,y0). 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. Ended with an explanation of the chain rule for finding the partial derivatives of composite functions.

Lecture 10
Proved the Fundamental Theorem of Calculus. Then started to introduce the concept of a differential 2-form. Got as far as talking about oriented planar triangles.

First Test (Monday 9th October 2017. The test is based on material up to, and including, the definition and calculation of the total derivative of a 1-form.)

Lecture 11
Explained what we mean by the integral of a constant 2-form over an oriented planar triangular region.

Lecture 12
Gave the definition of the integral of a (non-constant) 2-form over an oriented surface. Considered an easy example too.

Lecture 13
Gave an example of how to integrate a (non-constant) 2-form over an oriented surface. Then introduced the total derivative of a 1-form. Gave an example of how to compute the total derivative of a 1-form.

Lecture 14
Began by summarizing the rules needed to compute the total derivative of a 1-form. Then gave a second example of how to compute the total derivative of a 1-form. In Lecture 13 the rules of differentiation were used to calculate the total derivative of a certain 1-form. We finished the lecture by showing how this answer is precisely what is needed for Stokes' formula to hold for a 1-form (in two variables x,y over an oriented region S in the plane). This explanation provides some justification for our rules for differentiating 1-forms.

Lecture 15
Showed that under a mild hypotesis 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 kxk matrix).

Second Test

Lecture 16
Started with an illustration of how to find the total derivative of a 2-form. Then presented some general algebraic results on k-forms. Finished by calculating the area of a region in the plane bounded by a simple closed curve.

Lecture 17
Explained the the gradient of a function f(x,y,z) is essentially just the total derivative of the 0-form w=f(x,y,z). Then proved that the grad(f) is a vector normal to the surface f(x,y,z)=k with k a constant.

Lecture 18
Discussed the curl of a vector field.

Lecture 19
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 relavenca to you, then take a look at the Wikipedia page on econophysics.

Lecture 20
Explained the divergence of a vector field.

Third Test

Lecture 21
Missed a lecture due to Hurricane Ophelia!