The Arithmetic and Geometric Means You’re probably familiar with the arithmetic mean, which is most people mean (heh) when they say “average”: $$\displaystyle A(x_i) = \frac{1}{n}\sum_i x_i $$ where $x_i$ is a set of $n$ real numbers. The arithmetic mean answers the question “If all these numbers were equal, what would they have to be … Read More “Means and Concavity” »
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a circle, like this one: The center of this circle (the “circumscribed circle” or “circumcircle”) is called the “circumcenter”, marked by a dot in the above figure. Quadrilaterals have many special points that get called centers, such as the circumcenter, but the most … Read More “When is the centroid of a cyclic quadrilateral also its circumcenter?” »
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: $$\displaystyle \frac{d}{dt} \int_\Omega \rho\vec{u}\,dx = … Read More “Source of the source: Pressure in the axisymmetric momentum equation” »
Being almost obsessively parsimonious, I once joked that if everyone’s spending habits were like mine the economy would collapse. Mathematician that I am, I wondered if that was actually true and whether it could be predicted mathematically. Now you can enjoy the result. For simplicity, consider an economy with two kinds of people who differ … Read More “Power to the Penny-Pinchers” »
Suppose you have a set $S$ of $N$ numbers: $$\displaystyle S = \{x_1, x_2, x_3, \cdots, x_N\}$$ How many different products can be formed by multiplying at most $F$ members of $S$, allowing repetitions? For example, take $S = \{a,b,c,d\}$. Then the distinct products are: Number of factors Products 1 $$\displaystyle a, b, c, d$$ … Read More “Counting Products When Factors Count” »
1. Introduction One might remember from high-school precalculus class something called the Law of Sines, and might even remember what it is because it has a memorable pattern: $ \displaystyle \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} \ \ \ \ \ (1)$ where $a$, $b$, and $c$ (lowercase) are the side lengths and $A$, $B$, and … Read More “Law of Sines for Tetrahedra” »
The blog has set sail! This patch of the internet will be devoted to math that I, the Math Fish, find interesting. $\pi = 3.14159265358979…$