# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

## Featured Items

##### Means to an End

The limit of the geometric mean of the first $n$ integers raised to the real positive power $s$, divided by their arithmetic mean is shown to be $(s+1)/e^s$. An elementary derivation of Stirling`s approximation suggested this limit for $s=1$.

##### Yet Another Elementary Solution of the Brachistochrone Problem

The paper presents an elementary solution to the Brachistochrone problem.

##### Math Bite: $Q$ Is Not Complete

A series of rational numbers whose absolute values converge to a rational number while the series itself converges to an irrational number is presented.

##### A Direct Proof That Row Rank Equals Column Rank

A row (column) of a matrix is called “extraneous” if it is a linear combination of the other rows (columns).  The author shows that deleting an extraneous row or column of a matrix does not affect the row rank or column rank of a matrix.  This fact establishes the theorem in the title.

##### Finding Curves with Computable Arc Length

The author describes a method for identifying curves where the arc length is easy to compute by symbolic integration.

##### Fields for Which the Principal Axis Theorem Is Valid

One version of the Principal Axis Theorem asserts that any symmetric matrix with entries in $\mathcal{R}$ is similar over $\mathcal{R}$ to a diagonal matrix.  The authors find necessary and sufficient conditions for a field $K$ that make the Principal Axis Theorem valid over $K$.