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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” »
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” »
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” »
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?” »
You know about electricity, and you know about magnetism, and you’ve probably heard the word “electromagnetism” somewhere before to refer to electricity and magnetism combined in some sense. But why do these two seemingly different phenomena get lumped together into one word, whereas there’s no such thing (yet?) as “electrogravity”? If you know some more … Read More “Magnetism from Relativity” »
A sultan has granted a commoner a chance to marry one of his $N$ daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter’s dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must … Read More “The Sultan’s Daughters Problem” »
Several competitive decks in Vintage Magic: The Gathering are powered by the card Bazaar of Baghdad. Deck construction rules mandate at least 60 total cards in a deck with at most four copies of any given card. The game starts by each player drawing seven cards then performing, if he wishes, a series of “mulligans” … Read More “Mulligans in the Bazaar” »
In celebration of the unofficial end of summer (Labor Day in the U.S.), consider a snowpile on a trash bin: Observe how the shape of the lid (straight edges, rounded corners) seems to propagate upward, and how the sides of each pile are nearly planar with nearly the same slope on each side. Is this … Read More “Snowpile Math” »