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
However, my presentation of material in lectures is very much based on
Harold M Edwards advanced level book: Advanced Calculus: A differential forms
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
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
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.
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
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
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
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.
MA2286 module aims to give you the ability to benefit from
such research articles and books in financial mathematics!
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
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.
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
phrased in the language of differential forms. This problem will be finished
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.
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.
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.
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.
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.
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.
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.
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.)
Explained what we mean by the integral of a
constant 2-form over an oriented planar triangular region.
Gave the definition of the integral of a (non-constant) 2-form over an oriented surface. Considered an easy example too.
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.
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.
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).
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.
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.
Discussed the curl of a vector field.
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
Explained the divergence of a vector field.
Missed a lecture due to Hurricane Ophelia!