Have you ever wondered how famous results were actually done? If so, read on.
1. 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
2. Einstein's development of special relativity in his 1905 paper. Understand why his approach is not followed today. To read the paper click here: Special Relativity-how Einstein actually did it.pdf
3. How Maxwell derived his velocity distribution law. How did Maxwell derive his famous velocity distribution law, one of the foundational elements of statistical mechanics? As you will see, he used geometry and functional equation concepts to derive the law in only a few lines. To see how the gaps are filled in read the detailed paper here.
4. How the Stefan-Boltzmann law is derived from Planck's radiation law.
It is a standard homework problem in undergraduate physics to show how the Stefan-Boltzmann law can be derived from Planck’s radiation law. Underpinning the derivation one has to evaluate a certain type of integral and in this article I go through all the analytical steps involved in the evaluation. You need to know some analysis, Fourier theory and the properties of the gamma function. I also provide some historical information how Planck derived his law. Download the article here.
5. In 1905 Einstein produced a remarkable paper on Brownian motion. In this paper he derived from meagre physical and probabilistic assumptions a partial differential equation for the heat equation which had a well known solution at the time. I have gone through his succinct derivation in detail in this article.
6. The geodesic equation is a rite of passage for general relativity. Its derivation can be approached in various ways but it is possible to do a basic “bare hands” derivation which has been done for instance by Alan Guth in his MIT “Early Universe” course which can be found here: ocw.mit.edu/courses/8-286-the-early-universe-fall-2013/resources/lecture-14-the-geodesic-equation/ Guth leaves several important gaps in the derivation (because they would have been covered in tutorials or homework) but a complete development of the derivation is set out in the this paper. Because of all the indices there may be some typos despite several proof readings.