Posts with moderate math
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) $$ This formula uses two … Read More “Central Difference Formulas” »
Consider the equation for conservation of momentum in an inviscid flow, first in differential form: $$\displaystyle \frac{\partial}{\partial t} (\rho\vec{u}) = -\nabla\cdot(\rho\vec{u}\vec{u}) – \nabla P $$ and then integrated over an arbitrary control volume $\Omega$, using the divergence theorem on the momentum term and the gradient theorem on the pressure term: $$\displaystyle \frac{d}{dt} \int_\Omega \rho\vec{u}\,dx = … Read More “Source of the source: Pressure in the axisymmetric momentum equation” »
Being almost obsessively parsimonious, I once joked that if everyone’s spending habits were like mine the economy would collapse. Mathematician that I am, I wondered if that was actually true and whether it could be predicted mathematically. Now you can enjoy the result. For simplicity, consider an economy with two kinds of people who differ … Read More “Power to the Penny-Pinchers” »