Most easily digestible posts
Ridders’ method is a single-variable root-finding method that is more efficient than the basic regula falsi method. The formula is easy enough to find online (e.g. the first link), but its derivation is not. This post fills in the blanks. Suppose the function $f$ has a zero between $x_L$ and $x_R$, i.e. $f(x_L)f(x_R) < 0$. … Read More “Notes on the Derivation of Ridders’ Method” »
In multivariable calculus courses, one usually first encounters the divergence as: $$\displaystyle \nabla\cdot\vec{u} = \frac{\partial{u_x}}{\partial x} + \frac{\partial{u_y}}{\partial y} + \frac{\partial{u_z}}{\partial z} $$ where $\vec{u} = [u_x,\,u_y,\,u_z]$ in cartesian coordinates. Then one learns the divergence theorem: $$\displaystyle \int_\Omega \nabla\cdot\vec{u}\, dx = \int_{\partial\Omega} \vec{u}\cdot\hat{n}\,dS $$ and then, since cylindrical coordinates have usually already been introduced, one … Read More “Divergence in Cylindrical Coordinates – The Right Way” »
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?” »
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” »