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” »
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” »
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?” »
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” »