
Courses
Courses
Choosing a course is one of the most important decisions you'll ever make! View our courses and see what our students and lecturers have to say about the courses you are interested in at the links below.

University Life
University Life
Each year more than 4,000 choose NUI Galway as their University of choice. Find out what life at NUI Galway is all about here.

About NUI Galway
About NUI Galway
Since 1845, NUI Galway has been sharing the highest quality teaching and research with Ireland and the world. Find out what makes our University so special – from our distinguished history to the latest news and campus developments.

Colleges & Schools
Colleges & Schools
NUI Galway has earned international recognition as a researchled university with a commitment to top quality teaching across a range of key areas of expertise.

Research
Research
NUI Galway’s vibrant research community take on some of the most pressing challenges of our times.

Business & Industry
Guiding Breakthrough Research at NUI Galway
We explore and facilitate commercial opportunities for the research community at NUI Galway, as well as facilitating industry partnership.

Alumni, Friends & Supporters
Alumni, Friends & Supporters
There are over 90,000 NUI Galway graduates Worldwide, connect with us and tap into the online community.

Community Engagement
Community Engagement
At NUI Galway, we believe that the best learning takes place when you apply what you learn in a real world context. That's why many of our courses include work placements or community projects.
MA2286 Differential Forms
What will I study in this module?
ω is a differential pform in nvariables;
S is a "nice" (n+1)dimensional region in Euclidean space ℝ^{n+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 inclass 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 inclass tests contributes 10%.
Live lectures, oncampus 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 oncampus 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 0form in one variable. Differential forms are defined with respect to some nice oriented region S in ℝ^{n}. 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 0form in 1 variable over the boundary of a 1dimensional oriented region in ℝ. 
2 
Video of the lecture. I explained what is meant by a differential 1form in one variable. I also explained that the integral of such a 1form 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 1forms were used to model the rate of expendidture and the rate of income in a certain fund raising project. The integral of a differential 1form 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 0form 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 restatement 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 0form and differential 1form 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 1forms will be given next lecture. Also gave the definition of the derivative of a function f:R^{n} >R^{m}. 
5 
Video of lecture. Used the example of a particle in a constant force field to illustrate a (constant) differential 1form. Then used the marginal costs of acquiring/relinquishing assets from an investment portfolio to illustrate a (constant) differential 1form. Went on to explain how a differential 1form can be viewed as an infinite collection of vectors, one vector associated to each point in space. Finished by introducing the integral of a 1form on a connected, oriented 1dimensional 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 1form w along an oriented connected 1dimensional 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 1form. 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 1form. 
7 
Video of the lecture. Computed the work done in moving a particle once around an ellipse, anticlockwise, in a force field defined by a differential 1form on 3dimensional space. In the computation we used: (i) the definition of an integral of a 1form to simplify the problem to one involving a 1form 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 0form in several variables. So we now understand both sides of Stokes' formula for a 0form 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 0forms (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 1forms. 
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 (x_{o},y_{o}) if f(x_{o},y_{o}) equals the limit of f(x,y) as (x,y) > (x_{o},y_{o}). 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 2form 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 2forms over oriented planar triangles in 3dimensional space. I then started to give the definition of the integral of a (not necessarily constant) 2form over an oriented surface in 3dimensional 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 2form over an arbitrary oriented surface S. To reinforce our understanding of this definition a couple of integrals were evaluated. It is worth repeating: differential 2forms 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 2form represents, and we'll not worry too much about what a 2form itself represents. The algebraic rules for 2forms 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 2form over a planar region. Then explained the rules for differentiating a differential 1form w in order to obtain a 2form dw. I'm taking the view that: (i) differential nforms 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 nforms are; (iv) I'll be very precise in lectures about what is meant by the integral of an nform over an oriented ndimensional region; (v) in order to calculate integrals of nforms we'll need to know the rules for manipulating and differentiating nforms; (vi) these manipulation rules are forced on us by our understanding of an integral of an nform and by the desire to have Stoke's formula hold. So at the end of the module we'll be able to integrate over ndimensional regions using Elie Cartan's language of differential forms, a language that is perfect for: 
14 
Video of the lecture.
Provided some justification for the rules for differentiating 1forms w by showing that they suffice to establish Stokes' formula (for w a 1form in two variables x,y and S an oriented region in the xyplane). 
15 
Video of the lecture. Showed that under a mild hypothesis on a differential 1form w we have d(dw)=0. Gave an example which used this equation. Then quickly covered the basics on differential 3forms, and their integration, and the total derivative of a 2form. The treatment was such that it equally applies to differential kforms (though the 1dimensional notion of "length", and the 2dimensional notion of "area", and the 3dimensional notion of "volume" has to be replaced by the kdimensional notion of the determinant of a k×k matrix). 
16 
Video of the lecture.
I explained that a differential 0form w(x,y,z) in 3 variables, in conjunction with some constant k, can be thought of as the surface S defined by 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 0form 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 1form. Gave an interpretation of the curl (without proof). 
18 
Video of the lecture. Defined and interpreted the divergence of a vector field on 3dimensional 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 1form H, a 2form D and a 2form E^dt+B and a 3form 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.