## 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. 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. 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. 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. 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. 9 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. Ended with an explanation of the chain rule for finding the partial derivatives of composite functions. 10 First class test. It will consist of five questions from the problem sheet, up to and including Section 7. 11 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. 12 Explained what we mean by the integral of a constant 2-form over an oriented planar triangular region. 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. This lecture was delivered by Emil Skoldberg. 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. 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). 16 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. 17 Introduced the curl of a vector field, and gave an interpretation. Essentially: The gradient of a function f is got the derivative df where f is regarded as a 0-form. The curl of a vector field F is got from the derivative dF where F is regarded as a 1-form. Gave an example where I determined the equation of a tangent plane to a surface at a given point on the surface. 18 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. 19 Introduced the divergence Div(F) of a vector field F. Explained how Green's Theorem in the plane can be written as Stokes' formula for p=1 and two variables x,y and a simply connected region S in the plane. Explained how Stokes' Theorem from "Advanced Calculus" can be written as Stokes' formula for p=1 and a simply connected region S on a surface. Explained how the Divergence Theorem can be written as Stokes' formula for p=2 and three variables x,y,z and a region S on a 3-dimensional Euclidean space.. 20 Verified Green's Theorem in the plane on a particulare example. Then derived a general formula and used it to calculate the area of an ellipse. 21 The third class test will take place on Wednesday 28 November. 22 The second class test took place on Wednesday 14 November. 23 24