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
5. 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: Laplacian.mp4
12. Feynman developed his path integral approach to quantum physics in his PhD thesis and later he and Albert Hibbs produced a textbook on path integrals (“Quantum Mechanics and Path Integrals”, Emended by Daniel F. Styer ). In this book, Feynman and Hibbs pose several foundational problems which relate to the principle of least action. Full solutions to these problems ( Problems 2-1 to 2-5) can be downloaded here:
Solutions to Feynman-Hibbs classical action problems.pdf
13. I don’t normally publish solutions to problems (because there is a large industry of people doing this and I have much more interesting fish to fry ) but the following one from a 1985 Putnam contest piqued my interest because it struck me that it had to have a straightforward solution but still be worthy of Putnam difficulty.
View the solution here
14. The Koide formula in particle physics is a fascinating experimental proposition that may suggest deeper things. In this article I explore some of the maths. Download here.
15. I recently noticed an integration substitution trick in an a Youtube video https://www.youtube.com/watch?v=BfZObnTIsYk&t=110s that apparently came from an Indian high school exam. What was interesting was reverse engineering the trick (which was actually unnecessary anyway) and then showing how powerful simple substitutions are for the integral form of Bessel functions. View the article here.
16. Bessel functions are usually introduced in undergraduate Fourier analysis or engineering courses in the context of hanging chains and vibrating circular membranes, for example. However, the integral form of Bessel’s function actually arose from Bessel’s analysis of the eccentric anomaly in elliptic planetary motion and yet a modified form of Bessel’s function, known as a spherical Bessel function figures in the solution of a certain radial equation derived from the Schrodinger wave equation. To learn more see this article: Spherical Bessel functions in quantum mechanics.pdf
17. I have done a lengthy paper on the basics of Bessel functions which explains the historical genesis of them in the context of astronomy as well as giving many very detailed proofs covering equivalence of various forms of the functions, uniform convergence, analyticity and so on. It is an understatement to say that I have only touched the surface in what is a 66 page paper. Anyone who has ever read Watson’s tome on the subject will understand just how sprawling the subject is. To access the paper click here.