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
Imagine you have a fluid inside a cylinder that suddenly starts rotating at angular velocity
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” »
In multivariable calculus courses, one usually first encounters the divergence as:
The Arithmetic and Geometric Means You’re probably familiar with the arithmetic mean, which is most people mean (heh) when they say “average”:
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:
Consider the equation for conservation of momentum in an inviscid flow, first in differential form:
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