Maths at Bondi Beach


1. A piratical solution to a pursuit curve problem.  A fully worked calculus problem involving a pirate ship hunting a treasure ship. Download here

2. Something for high school students who find calculus mysterious.  Here is simple derivation of the area of a circle using an approximation by triangles: Area of a circle - limit of triangles

3. Famous Russian mathematician V I Arnold (died 2010) gave a speech in 1997 in which he made some very critical comments about what he perceived to be the state of French mathematics. Read the attached article to see what the issue was: V I Arnold's challenge to French mathematicians.pdf    

4. The equivalence of the algebraic and geometric forms of the dot product can easily be established from first principles given an understanding of direction cosines.  To see how read this article: 

Algebraic and geometric equivalence of the dot product.pdf

If you are looking for a  short “brute force”  proof of Jacobi’s Identity, go no further! Most textbooks either avoid the details or introduce a more sophisticated form of proof.  In reality it is possible to do a “bare hands” proof that is quite easy to follow. Download the paper here:

A short brute force proof of Jacobi's Identity.pdf

6. This short article gives detailed calculations for some basic angular momentum relationships in spherical coordinates as used in quantum physics. Angular momentum in spherical coordinates.pdf

7.  While reading a paper on celestial mechanics by Australian defence scientist Don Koks, I had occasion to brush up on some basic principles of celestial mechanics which led to a simple but fundamental observation about swept areas.  To read more download the short paper:  Celestial mechanics and circular reasoning.pdf

8.  Jacobi’s formula for the derivative of a matrix can be proved various ways but the attached paper uses basic linear algebra definitions to slowly illuminate the proof.  Download the paper here:  Jacobi’s formula for the derivative of a determinant.pdf    

9. A “bare hands” proof of Lagrange’s 4 vector cross product identity is not hard, just tedious.  You do it only once in your life but if you have trouble with doing it, read the attached paper: Lagrange's vector cross product identity.pdf


10.  For those who have forgotten or never knew how to work out square roots (and other roots) in their heads using Taylor's theorem, here is a reminder:  squareroots

11.  The Laplacian is the workhorse of physics and in the latest paper I have “kitchen sinked” the ways of expressing it in various coordinate systems with detailed calculations employing several different techniques. If you have trouble remembering how to work out the Laplacian in spherical coordinates, for instance, the paper specifically shows how you can remember the correct form. Download the paper here: The Laplacian in curvilinear coordinates - the full story

Download the tutorial on how to remember  the form of the Laplacian in orthogonal curvilinear coordinates: