Posts with moderate math
Benford’s law is an observation about the leading digits in sets of numbers that span a wide enough range and which arise “naturally”. Its most well-known claim is that the first digit tends to be 1 about 30% of the time instead of the 11.1% (1 out of 9) that one would expect. Benford’s law … Read More “What is Benford’s Law?” »
I recently upgraded from an old phone with a 3.5mm headphone jack to one that’s too new and fancy for such antiquated features. This created a nuisance because I had only wired 3.5mm headphones available, so instead of using my phone as an MP3 player I resorted to a dedicated MP3 player from the 2010s … Read More “Randomness on Repeat” »
In a previous post I showed that for any closed region $\Omega$, the integral of the outward-pointing surface-normal unit vector $\hat{n}$ over its boundary is zero: $$\displaystyle \int_{\partial\Omega} \hat{n}\,dS = \int_\Omega \nabla 1 \,dx = 0 $$ This post will explore a counterintuitive consequence of this fact. Suppose instead that we want the same integral, … Read More “Integral of the Unit Normal Over a Surface” »
Every whole number (except 1) can be factorized in one and only one way into a product of prime numbers. For example: $$\displaystyle \begin{aligned} 21 &= 3\times 7 \\ 34 &= 2\times 17 \\ 35 &= 5\times 7 \end{aligned} $$ By looking at the factorizations one can easily tell when two numbers share a common … Read More “Numbers With Nothing In Common” »
Consider a family of polynomials $\{P_0, P_1, P_2, \cdots\}$ where each $P_n$ has degree $n$. The polynomials are orthogonal if there is an inner product such that $(P_i, P_j) = 0$ if and only if $i\neq j$. Now, any given $P_{n+1}$ can be written in terms of all the lower-degree polynomials: $$\displaystyle P_{n+1} = r_{n+1,n+1}xP_n … Read More “Recurrence Relations for Some Orthogonal Polynomials” »
$\pi$ is the most well-known special number, but there is also $e$. Whereas $\pi$ has the easy interpretation of being the ratio of a circle’s circumference to its diameter, and everyone knows what a circle is, $e$ arises in calculus with which many people have no familiarity. Here’s something interesting… Without calculus, the closest one … Read More “What is e?” »
A previous post used the typical series-expansion-plus-linear-algebra approach for finding finite-difference formulas to derive approximations to the first derivative of any desired order of accuracy. If you’ve ever used that method yourself, you probably know how tedious it is, and that post doesn’t make it look much less so. On top of that, for all … Read More “All Finite-Difference Formulas for All Derivatives” »
Imagine you have a fluid inside a cylinder that suddenly starts rotating at angular velocity $\omega$: The fluid is initially stationary but eventually settles to a flow field that does not change further with time. What is the velocity of the fluid in the meantime? This question can be answered using Bessel functions. Simplifying the … Read More “Transient Flow in a Rotating Cylinder” »
While a student I often encountered sources claiming something to the effect of “no other special functions have received such detailed treatment […] as the Bessel functions”, which struck me as odd because in the undergraduate differential equations courses I took and taught no mention whatsoever was ever made of them, though the textbook might … Read More “Bessel Functions” »
The first derivative of an analytic function can be estimated by: $$\displaystyle h\frac{df}{dx}\Bigg|_{x=0} \approx -\frac{1}{2}f(-h) + \frac{1}{2}f(h) $$ where $h$ is a small distance. This is the common central difference formula; a more accurate but less frequently seen one is: $$\displaystyle h\frac{df}{dx}\Bigg|_{x=0} \approx \frac{1}{12}f(-2h) – \frac{2}{3}f(-h) + \frac{2}{3}f(h) – \frac{1}{12}f(2h) $$ This formula uses two … Read More “Central Difference Formulas” »