Special integrals: M-dimensional Gaussian

[latexpage] Regular readers of this blog will know that I love my integrals. For those that share the same enjoyment, I recommend this channel that often covers integrals from the MIT integration bee. For example, see this case of writing an integral in terms of itself. Very cool. In this post, I want to think …

Simply beautiful: Finding the covariant basis from seemingly nothing

[latexpage] Seeing how I wrote a new post yesterday in the area of tensor analysis, I was reminded of the beautiful result below. It likely would have made it onto this blog at some point, so I thought I would quickly write it all out. Here we go. *** Imagine we are presented with some …

Understanding metric elements for cylindrical coordinates

[latexpage] The metric tensor is ubiquitous when arriving at a certain level in one's physics career. When it comes to cyndrilical coordinates, there is a useful way to remember its deeper meaning through a rather simple derivation - or at least through the use and construction of a series of definitions. (I say 'simple' in …

Leibniz Rule: Differentiating under the integral sign (special case)

[latexpage] I recently wrote a short piece on the general case of the Leibniz Rule. In that article I alluded to how there is also a special case of the rule, in which the limits of integration are constants as opposed to differentiable functions. In short, the special case of the Leibniz Rule is a …

Leibniz Rule: Differentiating under the integral (general case)

[latexpage] Evaluating integrals has over time become one of my favourite activities. Whether it is by parts or by substitution or by partial fractions or reduction - there are many cool integrals to be found in the world of applied maths. Two especially cool methods for solving definite integrals are the Leibniz Rule (or what some …

x^x^x^2017=2017 – A fun problem with a unique and sad history

[latexpage] I was sent this problem a few weeks ago by a physicist who I work with privately on a weekly basis (on the side of my school studies). I found it to be a lot of fun to think about, and was eager to write about the solution. Before getting into the solution, from …

The Road to Reality

I have been reading, working through and thinking a lot about Roger Penrose's magnum opus, The Road to Reality: A Complete Guide to the Laws of the Universe. It is a book that I cannot speak more highly about, possible one of the best books that I have read so far in my lifetime. My …

Mass and energy: From particle accelerators to thinking how much energy is stored in a mass of a glass of water

[latexpage] The title of this article refers to a fun example that pertains to the relationship between mass and energy. But before we get to that, some definition is required. For the sake of general readership, I will try to keep things simple. To start, we need to get into some physics. In particular, we …

How I taught myself calculus – Some intuitive reasoning

When I first taught myself calculus prior to more formal learning, I already had an idea of the concept of differentiation as a tool to help us measure the rate of change of a curve. I also already had some idea of higher differentials, a vague sense of integration, and some basic suspicions with regards to the …

Why 0!=1 – Talking math and terminal access code decryption in No Man’s Sky

Several months ago I made a video on a particular terminal access code decrypt problem that I had come across in No Man's Sky (see above). The given sequence was 1 2 6 24 120. In order to decrypt the terminal and retrieve the valuable information, it was my job in the game to find …