Maths at Bondi Beach


Mathematics without the mystery. Try it for yourself!


  • NEW !!  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

  • As part of a new section titled “Famous proofs - how they actually did the business” I have provided a detailed explanation of how in his 1905 paper Einstein developed the transformation equations at the heart of special relativity.  Einstein never returned to the form of explanation he used in the 1905 paper and, after reading the paper, you may appreciate why the subject is taught in a different way today.  To download the paper click here: Special Relativity-how Einstein actually did it.pdf
Einstein’s 1905 paper can be accessed here
  • I have substantially expanded the existing paper on the wave equation and energy conservation to cover the general case of d dimensions where d > 1.This involves a level of mathematical sophistication far greater than the one dimensional case dealt with initially.  A fundamental part of the derivation involves differentiation under the integral sign, which requires a detailed discussion of the d dimensional Leibniz rule.  The Reynolds Transport Equation is effectively this rule and there is a detailed discussion of Harley Flanders’ reconciliation of mathematicians’ and physicists’ proofs of the Leibniz rule.  You are warned – this is not for the faint hearted but I have put in all the detail so you should not get lost. Download the new paper here: The wave equation and energy conservation.pdf
  • 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 hereJacobi’s formula for the derivative of a determinant.pdf
  • 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 paperCelestial mechanics and circular reasoning.pdf

  • Tutorial on the uniform continuity of the Fourier Transform - Parts 1 and 2.  If you want to understand why, using classical analysis techniques, the Fourier transform of a function is uniformly continuous read this paper and watch the accompanying videos.  The  two part tutorial can be viewed here:
or downloaded in lower definition (.wmv files) here:

Part 1
The accompanying paper can be downloaded here  
  • Rademacher functions are simple step-functions which have a surprising role in the concept of statistical independence.  This role was explored by Mark Kac in his well known book “Statistical Independence in Probability, Analysis and Number Theory”.  One of the basic properties of Rademacher functions is their orthogonality (actually orthonormality) and in this paper I demonstrate a purely combinatorial proof of the orthogonality and contrast it with the traditional proofs.  To view the paper click A combinatorial view of the orthogonality of Rademacher functions.pdf

  • The Fejer kernel figures prominently in the theory of convergence of Fourier series and, unlike the Dirichlet kernel, it is well behaved.  This good behaviour is explained by the Fejer kernel’s use of Cesaro sums.  My detailed paper sets out all the relevant derivations (with multiple styles of proof for the most important results). If you are doing some serious Fourier analysis this will be of interest. To download click  here: The nitty gritty of Fejer's Theorem.pdf

  • 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
  • As part of a new section titled “Famous proofs - how they actually did the business” I have provided a detailed explanation of Laplace’s original proof of the Central Limit Theorem.  To read Laplace’s proof click here:The central limit theorem - how Laplace actually proved it.pdf
  • Laplace’s method of estimating the leading order behaviour of certain integrals is a powerful technique which can be used to prove Stirling’s formula among many other things.  The proof of Laplace’s method is a problem in Polya’s and Szego’s famous book “Problems and Theorems in Analysis 1”.  Their proof is rigorous but skips many fine details (which is not surprising given the level at which the book is pitched) which I have filled in in the download: Laplace’s method for integral asymptotics.pdf

  • Problem 1017 in the January 2015 College Mathematics Journal (Vol 46 No1 ) gives a very slick solution to a matrix problem.  Unfortunately there are some typographical errors in the published proof but these are fixed in the attached note which expands on the basic theory used in the solution. Simplifying a matrix expression.pdf
  • Australian mathematician/statistician Peter Donnelly who works at Cambridge University demonstrates in a TED talk (  )  just how poorly the judicial system handles probabilistic and statistical arguments.  He does this by posing a simple coin tossing experiment and then develops how the English  court system put a mother in gaol for the alleged murder of her two children because of some faulty statistical arguments.  To understand how juries  and judges can be fooled by statistics clickFooling juries with statistics.pdf 
  • While soft sand running with Bondi Beach’s surfing physicist Ruben Meerman  of the ABC’s “Catalyst” program we were discussing, among other things, why 1-1+1-…=1/2.  My paper on Cesaro summability explains what is going on with such series and how mathematicians, like Mussolini redefining lateness so the trains ran on time (undoubtedly a myth but it sounds good), redefine what sums of such series are so that you get nice behaviour. The basics of Cesaro summability
  • 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


  • 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


  • In his famous probability textbook, William Feller derived a surprising result in relation to the variance of n mutually independent random Bernoulli variables with variable probabilities of success. If we focus on Bernoulli trials with a constant probability of success which equals the average of the variable probabilities we get the counterintuitive result that the variability of the original probabilities, or lack of uniformity, decreases the magnitude of chance fluctuations as measured by the variance,  Thus given a certain average quality p of n machines, the output will be least uniform if all machines are equal. Think of how this principle applies to investment managers and MOOCs (massive on-line open courses).  To read the detailed argument and derivations click here: Bernoulli trials with variable probabilities - an observation by Feller.pdf
  • Ever wondered how the residents of Easter Island managed to get those huge stone heads in place? Well, that’s a bit like wondering how they solved quartic equations in the 16th century without computers.  To understand how it was done have a look at the  following paper which builds upon my paper on how to solve cubic equations (which you should read first or watch the video): Solving a quartic by the method of radicals.pdf
  • High school students might be interested in how to derive Cardano's formula for the cubic. A detailed paper is located here:  All you wanted to know about solving cubics but were afraid to ask
  • Youtube link:
  • While it is possible to apply Fourier theory without knowing precisely why all the integrals converge, for those who worry about such things or actually have to demonstrate some understanding in an analysis exam, my detailed paper on “Basic Fourier Integrals” may help. It expands material covered in Elias Stein and Rami Shakarchi's Princeton Lectures on Fourier Theory. The concept of Schwartz space is developed in detail against the background of older approaches. Applications of Fourier theory are explained and the use of Fourier transforms to solve the Black-Scholes equation from finance is done in great detail. Download the paper here: Basic Fourier integrals.pdf
  • 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

  • Because analysis is so highly proof driven students need to develop a facility with fundamental logical operations that underpin the proofs. In essence this boils down to understanding proof by contradiction and how to negate complex definitions such as the  limit concept, continuity, uniform continuity, differentiability and so on. For more download: Basic logic for first year analysis students.pdf
  • Chebyshev’s inequality with a hint of measure theory.  The Rigour Police will come around and beat me senseless for this but here goes anyway. Who knows, someone may find it useful.  Chebyshev
  • Proving the uniform continuity of sin x/x with some connections to Fourier transform theory. If you really want to get your hands dirty with epsilon-delta proofs go no further:  Uniform continuity of sinc x.pdf
  • When Richard Feynman and Richard Hibbs initially published their book “Quantum mechanics and path integrals” in 1965 they described a conceptual double slit experiment (see page 3 of the 1965 Dover edition)  and commented that “this particular experiment has never been done in just this way”.  When Daniel Steyer edited the book in 2005 he added footnotes pointing to a range of subsequent experiments which really did achieve what Feynman was describing in his conceptual treatment (see page 361 of the 2005 Dover edition). 
    This link is an example of the progress that has been made in the experimental world:
    double slit video
  • When Schwarz proved his famous inequality he used an insight that enables a one line proof. In addition I have included Cambridge mathematician Tadashi Tokieda's novel viscosity proof. To understand that proof  click here: A one line proof of the Cauchy-Schwarz inequality.pdf
  • If you want to brush up on some more advanced inequality techniques have a look at this paper which takes you through the solution process for some more difficult types of inequalities.  You will need a knowledge of the Arithmetic Mean –Geometric Mean inequality and the Cauchy-Schwarz in equality.  The Rearrangement Theorem and Chebyshev’s inequality are also covered.  Download here  
  • A piratical solution to a pursuit curve problem.  A fully worked calculus problem involving a pirate ship hunting a treasure ship. Download here
  •  Applying Riemannian integration theory to some practical examples.  This is the mathematical equivalent of building a pyramid with a trowel, but having done it once (and only once!) you will appreciate the power of the theory.  The article can be accessed here:  Riemann integration- some practical examples.pdf
  •  Dirichlet’s test for the convergence of series.  In this short article I fully develop the test which is useful for oscillatory series. There is a derivation of the summation by parts formula. Click here to download  the article:  Dirichlet’s test for convergence
  • If you are serious a student of analysis and want to know more about how some fundamental aspects of Fourier theory are proved in detail, why not have a look at my paper on kernels which is based on Elias Stein and Rami Shakarchi’s book: “Fourier Analysis: An Introduction”.  The title of the paper is: “The good, the bad and the ugly of kernels: Why the Dirichlet kernel is not a good kernel”.

    In the paper there is a wealth of detailed analytical proofs of fundamental issues that are exercises in the book eg detailed proofs relating to ordinary, Abel and Cesaro summability and the role of Tauberian conditions.

    I have not skipped steps in order not to lose those who have not immersed themselves in this topic.

     Click here to download:The good, the bad, and the ugly of kernels.

  • If you are having trouble understanding the binomial series for negative integral exponents you may be interested in this article which gives a simple but rigorous explanation of the concepts: The binomial series for negative integral exponents.pdf
  • I have uploaded a detailed paper on Laplace's Law of Succession ("What is the probability that the sun will rise tomorrow?"). This paper is based on Kai Lai Chung's rederivation of the relevant probability by using an urn model.  I have gone through all the intricate combinatorial steps so if you want to see all the gory details read on. A much simpler continuous derivation is also given. Laplace's Law of Succession.pdf
  • A bound for an integral: this problem has figured in various guises in a number of problems in the American Mathematical Monthly.  The original solution goes back to 1969 and is only 4 lines long, leaving out all the details.  The original problem  with a hint  and solution are here: A bound for an integral.pdf
  • If you are having trouble with integral reduction formulas I have done a short paper covering a representative set of examples.  If you are doing the NSW HSC these may assist.   Click here to get the paper :  HSC Integral Reduction Formulas.pdf

  •  Some years ago I picked up a standard high school textbook and randomly opened a page on trigonometry. There were many problems about ladders against walls. Nowhere could I find the one thing that for me is the hallmark of mathematical thinking - imagination.  
  • As part of an on-going experiment in maths education I am releasing extensive theoretical material coupled with problems and their solutions for interested teachers and students. The material is pitched at students who are curious about mathematics, yet perhaps fearful thanks to some teacher in an ill-fitting cardigan who humiliated them about quadratic equations. However, after a taste of some of its wonder, I hope that you might become passionate about one of the most sublime pleasures available.    
  • I was galvanised into doing this because of the parlous state of mathematics education at secondary and tertiary levels. Deeper laments can be found at, in particular Lockhart’s Lament. The full story behind my concerns can be found here:
  • I am not alone in my concerns.  The maths community in the United States spends an enormous amount of its time addressing the quality of content and new ways of delivering material.  Anyone seriously interested in maths should join the Mathematical Associaton of America
  • The theoretical material is meant to extend a student's range of knowledge and excite interest.  Some of the material may appear too difficult and that may well be the case, but there is a benefit for students in going with the flow and seeing how the bits fit together even if some of the detail eludes them. I have gone to great effort to put all the reasoning in so there aren’t any major gaps.  
  • The problems are designed to establish a number of basic competencies and there is a range of difficulty. 
  • By way of overview I have taken the most "boring" mathematical proof technique in the universe (ie mathematical induction) and shown how it figures in a wide range of significant problems. Feynman's path integral approach which he used to investigate quantum mechanical behaviour actually involved induction (admittedly in a minor way) and I show how it was used in the context of explaining why light travels in a straight line.  Along the way the basic physics are also explained.  Other applications in logic, probability theory and much more are also explained.  If you want to know whether you can prove by induction if it is possible to escape the Gulag of chartered accountancy, there is a problem on the very subject. It may come in handy depending on your career choice!
  • Even if you aren't interested in the theoretical material and you just want to do problems I will not be offended.  There are some interesting problems which bring together a range of different mathematical techniques which might assist in developing mathematical insights.  My proof of the volume of an n-dimensional ball using only elementary calculus and induction is an example of what can be done. 
  • This experiment is designed to get students interested in doing maths even though they may not have any intention of pursuing a highly mathematical career option. Because maths is an enabling discipline, the intellectual reach of other disciplines can be limited by lack of mathematical knowledge. Watson and Crick (and Ms Franklin whose role in the discovery also deserves recognition) used Fourier Theory to determine the structure of DNA.  They were molecular biologists/biochemists.  
  • I have made every attempt to ensure that the material is error free but in something of this scale errors will arise.  If you identify errors please email me with details via the contact page.    
  • To pique your interest in probability let’s start with Laplace’s famous problem concerning the rising of the sun.  If the sun has risen on n days in the past what is the probability it will rise tomorrow? Laplace’s perhaps counterintuitive answer was 2/3 notwithstanding that he assumed that the probability of  the sun rising on any particular day was ½.  The young Kai Lai Chung was puzzled by this and developed a combinatorial proof for this miraculous result.  Probability is full of such conundrums.  Kai Lai Chung went on to be a famous probability theorist.   


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