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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” »
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” »