Any rotation in 3D space can be viewed as three successive rotations about predefined axes, giving the Euler or Tait-Bryan angles). It can also be viewed as a pair of rotations, one about a predefined axis (the roll axis) and another that aligns the new roll axis to the original. In the aeronautical context, the … Read More “Swing-Twist Decomposition of a Quaternion” »
In fluid dynamics, the “strain rate tensor” is declared to be: $$\displaystyle S_{ij} = \frac12\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) $$ This quantity is usually derived using a diagram of a square fluid element in 2D and considering the rates of increase of two angles as the square deforms into a parallelogram. … Read More “The Strain Rate Tensor” »
Benford’s law is an observation about the leading digits in sets of numbers that span a wide enough range and which arise “naturally”. Its most well-known claim is that the first digit tends to be 1 about 30% of the time instead of the 11.1% (1 out of 9) that one would expect. Benford’s law … Read More “What is Benford’s Law?” »
I recently upgraded from an old phone with a 3.5mm headphone jack to one that’s too new and fancy for such antiquated features. This created a nuisance because I had only wired 3.5mm headphones available, so instead of using my phone as an MP3 player I resorted to a dedicated MP3 player from the 2010s … Read More “Randomness on Repeat” »
Given a real number $a$, the exponential $\exp(ta)$ is the solution to the initial value problem: $$\displaystyle \begin{cases} \frac{dy}{dt} = ay & \\ y(0) = 1& \end{cases} $$ This definition does not depend very much on the fact that $a$ is a real number. $a$ could perfectly well be a complex number and nothing in … Read More “Left and Right Exponentials” »
In a previous post I showed that for any closed region $\Omega$, the integral of the outward-pointing surface-normal unit vector $\hat{n}$ over its boundary is zero: $$\displaystyle \int_{\partial\Omega} \hat{n}\,dS = \int_\Omega \nabla 1 \,dx = 0 $$ This post will explore a counterintuitive consequence of this fact. Suppose instead that we want the same integral, … Read More “Integral of the Unit Normal Over a Surface” »
The Sherman-Morrison-Woodbury formula expresses the inverse of an “update” to a matrix $A$ in terms of $A^{-1}$ and the factors of the update: $$\displaystyle (A + UCV)^{-1} = A^{-1} – A^{-1} U\left( C^{-1} + VA^{-1}U \right)^{-1}VA^{-1} $$ where $A$ and $C$ are invertible square matrices and $U$ and $V$ have dimensions such that $UCV$ has … Read More “Sherman-Morrison-Woodbury for Determinants” »
Suppose we have a region $\Omega \subset \mathbb{R}^n$ that is a polytope, i.e. a non-empty intersection of half-spaces. Visually this would be a polygon in 2D or a polyhedron in 3D – “polytope” is a single term to describe such objects regardless of the number of dimensions. The important feature is that the boundary of … Read More “Polytope Identities from Vector Calculus” »
Any motion of a rigid object is equivalent to a rotation around some axis followed by a translation along that axis. We can therefore represent any motion mathematically using some object that includes information for a single rotation and a single translation. An extremely convenient object for this purpose is the dual quaternion. Part of … Read More “The Exponential and Logarithm for Dual Quaternions” »
Introduction In dynamics, one frequently needs a derivative of a vector in an inertial frame of reference in order to apply Newton’s laws when it is much more convenient to express that vector in a different frame. The Basic Kinematic Equation (also known by many other, less-informative names) relates the derivatives in two frames to … Read More “What is the Basic Kinematic Equation?” »