In this category you can find information regarding courses that I am teaching. For information about my teaching qualifications and philosophy, see the Professional category.

Since Autumn 2009, I have used a wiki for course notes which can be found at http://mathsnotes.math.ntnu.no. I also maintain a forum which is intended for students of the courses that I teach, this can be found at http://mattesnakk.mathforge.org. Both produce MathML by server-side conversion.

• Spring 2012: Matematikk 3, section FYMA
• Autumn 2011: TMA4145 Linear Methods
• Spring 2011: Matematikk 3, section FYMA, other information: TMA4115v2011
• Autumn 2010: TMA4145 Linear Methods
• Spring 2010: Matematikk 3, section DT, IØT, KOM
• Autumn 2009: TMA4145 Linear Methods
• Spring 2009: TMA4190 Manifolds
• Autumn 2008:
  • TMA4145 Linear Methods
  • TMA4170 Fourier Analysis
• Spring 2008: TMA4230 Functional Analysis
• Autumn 2007: MA3402 Analysis on Manifolds

The following are small programs that may (or may) not be useful. They are listed here as it is likely that their primary reason for existing is for use in teaching.

Some are Java applets (or standalone Java programs) developed using processing. These will work in a browser (that supports Java) or can be downloaded to a Linux, MacOSX, or Windows machine. Others are Lua scripts developed on the iPad application Codea. These are designed to be run within Codea on an iPad, though it may be possible to adapt them to run using Love2D (see this discussion for details). Getting code onto the iPad can be somewhat tricky, see this discussion for methods.

The Java applets were originally released under the GPL. The Codea code is released under the CC0 to the extent allowed (some pieces are code that I've adapted from others; see Codea for specific details).

• TMA4115.
  • Coupled Pendulum. Applet, standalone application for linux, MacOSX, Windows. Created using processing.
  • Dynamic Matrix. Applet, standalone application for linux, MacOSX, Windows. Created using processing.
  • Discrete Matrix. Applet, standalone application for linux, MacOSX, Windows. Created using processing.
  • Apply Matrix. Applet, standalone application for linux, MacOSX, Windows. Created using processing.

The most basic object of study in mathematics is of a process. Processes take in input and spew out output. Almost everything that one wants to study scientifically can be modelled mathematically by a process. Given a process, there are three types of question that one wants to answer:

1. What I put in X, what did I get out?
2. Where I got out Y, where did I start?
3. How I put in X, I got out Y, how did I get it?

We model processes in mathematics by functions. Thus the study of processes leads one, in mathematics, to the study of functions. In the first two questions, the function is (presumably) known and the questions are about studying it. The last question in the list above is concerned with the question of finding a function. To do this, one must have some idea of the types of function that might fit, how these behave, and how to describe them. This involves studying not just one function but whole families of functions.

One of the simplest such cases is the family of continuous functions on the interval. One can think of these as modelling processes whereby:

1. The input is a parameter between 0 and 1.
2. The output is a (real or complex) number.
3. The output depends continuously on the input.

In this course, we shall develop the tools necessary to study such functions and, more importantly, the space of such functions.

Linear problems occur throughout mathematics and its applications. This course will examine the mathematical tools needed to study such problems. The simplicity and elegance of these tools is part of what makes linearity such a desirable property. Indeed, many problems that are not inherently linear are often made linear as a first step in their analysis.

Problems that require only a finite number of pieces of information to specify them can be reduced to the study of matrices and coordinate vectors. However other problems, such as the evolution of a sound wave, require so-called infinite dimensional analysis. In this course we shall study both.

The topics that we shall cover include:

• Approximation and metric spaces
• Linear transformations and their characteristics
• Infinite dimensional linear spaces

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Manifolds occur throughout mathematics -- and hence other sciences -- as spaces in which something interesting happens or spaces in which something interesting is to be found. Often the set of solutions to some problem forms a manifold, or the set of possible configurations of some system forms a manifold. By studying manifolds and by developing a set of tools with which to study particular manifolds, we can gain considerable insight into those problems where manifolds occur.

The key property of a manifold is that it locally looks like ordinary Euclidean space. Therefore, things that work on small patches of Euclidean spaces can often be made to work on manifolds. The most important of these things is calculus. Indeed, one can regard manifolds as the right places to do calculus and many of the questions involving manifolds have their origin in considering a particular theorem of calculus.

This course is designed to be an introduction to the theory of manifolds, looking at particular examples and developing simple techniques for studying them.

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Linear problems occur throughout mathematics and its applications. This course will examine the mathematical tools needed to study such problems. The simplicity and elegance of these tools is part of what makes linearity such a desirable property. Indeed, many problems that are not inherently linear are often made linear as a first step in their analysis.

Problems that require only a finite number of pieces of information to specify them can be reduced to the study of matrices and coordinate vectors. However other problems, such as the evolution of a sound wave, require so-called infinite dimensional analysis. In this course we shall study both.

The topics that we shall cover include:

• Approximation and metric spaces
• Linear transformations and their characteristics
• Infinite dimensional linear spaces

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Fourier analysis involves rewriting functions of position as functions of frequency. In many situations, the behaviour of a system is much simpler when it is viewed as dependent on frequency rather than position. It is therefore a Good Thing to have the output of the system written in terms of frequency. However, it is often much easier to find the values of that system in terms of position. The techniques of Fourier theory allow us to make the best of both: we can read off the values in terms of position and then use Fourier analysis to rewrite these in terms of frequency.

In this course we shall be studying the mathematical tools and ideas behind Fourier theory.

This course studies the central ideas of functional analysis, focussing primarily on normed vector spaces.

Highlights include:

• The Hahn-Banach theorem (in various guises)
• The open mapping and closed graph theorems
• The Banach-Steinhaus theorem
• Dual spaces
• Weak convergence
• Banach-Alaoglus theorem
• Spectral theory for self-adjoint operators

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An introduction to the ideas, concepts, and techniques of analysis on manifolds.

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The most basic object of study in mathematics is of a process. Processes take in input and spew out output. Almost everything that one wants to study scientifically can be modelled mathematically by a process. Given a process, there are three types of question that one wants to answer:

1. What I put in X, what did I get out?
2. Where I got out Y, where did I start?
3. How I put in X, I got out Y, how did I get it?

We model processes in mathematics by functions. Thus the study of processes leads one, in mathematics, to the study of functions. In the first two questions, the function is (presumably) known and the questions are about studying it. The last question in the list above is concerned with the question of finding a function. To do this, one must have some idea of the types of function that might fit, how these behave, and how to describe them. This involves studying not just one function but whole families of functions.

One of the simplest such cases is the family of continuous functions on the interval. One can think of these as modelling processes whereby:

1. The input is a parameter between 0 and 1.
2. The output is a (real or complex) number.
3. The output depends continuously on the input.

In this course, we shall develop the tools necessary to study such functions and, more importantly, the space of such functions.

The lectures will be delivered as PDF presentations. I shall endeavour to make the notes available the day before the lecture so that students can "follow along". There may be additional notes written during the lectures. These will be posted shortly after the lecture finishes. I shall often prepare a little more than I shall actually give, any extra will be taken up in the next lecture or deferred to the wiki.

The notes will be available in several different layouts. It is important to know which is which.

1. Beamer. This is what will actually appear on the screen during the lectures. You must never print this version. As each "transition" results in a new page, this can easily exceed 100 pages.

2. Trans. This is a "one frame per page" version of the above. If you intend to "follow along" with the lecture on your own computer then this is probably the best. However, I still strongly urge you not to print it.

3. Handout. This is a more condensed version in terms of space. By putting 4 slides on a page the total number of pages is significantly reduced. If you want to print something then print this version.

4. Annotated. It will probably take a little experimenting to find the best way to present the annotated version. As a first go, the PDF contains just the annotated pages. For most pages, this should be sufficient to locate it within the main presentation. For some it may be useful to have the previous page included as well. Let me know if this, or something else, would be useful.

Note: all are as PDFs.

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1. 10th January. Complex Numbers. beamer trans handout annotations

2. 11th January. Complex Numbers and Powers. beamer trans handout annotations

3. 17th January. Differential Equations: Who? What? Why? Where? How? beamer trans handout annotations

4. 18th January. ODES: The Simplest Case. beamer trans handout annotations pendulum applet (see also applets)

5. 24th January. Set an ODE to Solve an ODE. beamer trans handout annotations

6. 25th January. Inhomogeneous ODEs. beamer trans handout annotations

7. 31st January. Lecture cancelled

8. 1st February. Resonance. beamer trans handout annotations pendulum applet (see also applets)

9. 7th February. Linear Systems. beamer trans handout annotations matrix applet (see also applets)

10. 8th February. Matrices and Linear Systems. beamer trans handout annotations

11. 14th February. Matrices. beamer trans handout annotations

12. 15th February. Invertibility. beamer trans handout annotations

13. 21st February. Inverting Matrices. beamer trans handout annotations

14. 22nd February. Muddiest Point. No advance notes. annotations

15. 28th February. Voluntary Midterm. No advance notes. annotations

16. 1st March. Vector Spaces. beamer trans handout annotations

17. 7th March. Describing a Subspace. beamer trans handout annotations

18. 8th March. Bases. beamer trans handout annotations

19. 14th March. Another Angle of Attack. beamer trans handout annotations

20. 15th March. Intrinsic Orthogonality. beamer trans handout annotations

21. 21st March. Decompositions. beamer trans handout annotations

22. 22nd March. Finding Eigenvectors. beamer trans handout annotations

23. 28th March. Things to Do With Eigenvectors. beamer trans handout annotations

24. 29th March. Muddiest Point. annotations

25. 4th April. Orthogonal Eigenvectors. beamer trans handout annotations

26. 5th April. Quadratic Forms. beamer trans handout annotations

27. 11th April. Revision Lecture: ODEs. annotations

28. 12th April. Revision Lecture: Linear Algebra. annotations

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