Have you finished your High School Mathematics curriculum early and want to be extended with more mathematics? Extension III mathematics is a self-paced course that may be of interest for you. It is designed specifically for those of you who have finished the Extension 2 syllabus here in New South Wales, Australia, but it is suitable for many talented high school students from different backgrounds. The name is partly a nod to the prestigious Part III of the Mathematical Tripos at the University of Cambridge, a famous training ground for young mathematicians wishing to pursue a PhD in mathematics.
The style of mathematical learning here is very different from what is found in schools or even in the first few years of undergraduate since it is designed to be explorative and encourage you to think independently. Instead of giving detailed recipes of mathematical methods and giving oodles of questions to
practice these methods, you work through Activity Sheets which encourage you to figure out interesting mathematical theory as you go along as well as plenty of interesting applications of that theory. Many Activity Sheets will also have a video attached to them that you can watch beforehand which serve as a teaser for the material. The activities are challenging since this course is designed for those who found the Extension 2 (or equivalent) curriculum easy. It is not just a continuation of Extension 2. Ideally, you should have a mentor to help you along. If you completed Extension 2, you can contact
Professor Daniel Chan to see if you qualify for the Extension III program at the School of Mathematics and Statistics at UNSW.
The mathematical content is designed not to replicate university mathematics, but to complement it by exploring important mathematical topics which unfortunately often do not make it to standard curricula. Bits and pieces of the standard university Calculus and Linear Algebra material will make their appearance, but in a way that hopefully will make you better appreciate them when you get to university.
Enjoy!
Construction warning This website is still work in progress and in particular, videos will be
added, hopefully on a weekly basis.
Activity Sheets and Videos
Below are the Activity Sheets and links to videos. The Activity Sheets include both activites as well as many partial answers. Make sure you read all the latter as they include a lot of information useful for later Activity Sheets and give you insight into the material. They are written in a particular order, but you can do some out of order or even skip some. Look in the Activity Sheets themselves if you wish to do this.
Topic: Generating Functions in Enumerative Combinatorics
Many problems in mathematics involve counting objects, for example, the number of permutations of
n symbols is n!. Such problems are a part of a branch of mathematics called
enumerative combinatorics. A rather interesting way to study such problems is by through the use of
generating functions, where now the calculus of functions can be brought to bear on counting problems. The
Activity Sheets in this topic will dip in to the calculus you need to know (infinite sums and power series) and then see how to study numerous interesting counting problems.
Watch the video above for a teaser on the notion of recurrence relations and generating functions. Then try the Activity Sheet
Recurrence Relations and Generating Functions to see if you can use generating functions to encode and solve recurrence relations.
Watch the videos above to learn about the following concepts from calculus: formal definition of limits
of sequences, the Cauchy completeness theorem for existence of limits, Cauchy products of series and the ratio test. Then try the Activity Sheet on Series to gain some familiarity with these important concepts from
calculus.
Watch the videos above on Power Series and Taylor's theorem. Try the Activity Sheet on
Power Series, an indispensable tool for understanding many functions. This also legitimates the manipulations one performs with generating functions.
Watch the teaser video above on Counting Monomials (make sure you know the binomial series from the Power Series Activity Sheet before doing so). Then try the Activity Sheet on
Counting Monomials to see if you can solve various related counting problems using monomials and the resulting generating functions.
You might think that one shouldn't have to use calculus techniques to study counting problems, and you're right. However, to replace the theory of Maclaurin series, you need to look at Formal Power Series to legitimise all your manipulations of generating functions. Try your hand at the Activity Sheet which sets up the theory and introduces the modern algebraic notions of rings and fields.
Watch the video above for a teaser on Catalan numbers arising in the bracket counting problem. You'll need the binomial series from the Activity Sheet on Power Series to follow it. Then try your hand on the Activity Sheet
Catalan Numbers to see various other counting problems where they arise.
You've studied many counting problems, but what does counting really mean? Watch the video above to learn some of the basic language of bijections, which forms the mathematical basis of the notion of counting. Afterwards, try the Activity Sheets Counting and Bijections.
Information about generating functions is often encoded in differential equations they satisfy. Watch the video to see how to solve differential equations which are first order and linear. Then try your hand at the Activity Sheet
Linear Ordinary Differential Equations.
Sometimes it is good to use a variant of the usual generating function called the exponential generating function. Watch the video above to see an example studying derangements. Afterwards, you can try your hand at the Activity Sheet Exponential Generating Functions.
One of the most interesting enumerative problems involves counting various partitions. If you don't know what partitions are, look at the teaser video above (coming soon) which will also tell you how to use generating functions to count them. Then try your hand at the Activity Sheet Partitions.
There is a discrete analogue of the differentiation operator called the difference operator
which is useful for studying sequences in much the same way that the usual differentiation operator is useful for studying functions. Check out the teaser video above to see how and then try your hand at the Activity Sheet
Difference Operator.