# 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

##### How Commutative Are Direct Products of Dihedral Groups?

Given a positive integer $m$, the authors exhibit a group with the probability $1/m$ that two randomly selected elements commute.

##### Solving the Noneuclidean Uniform Circular Motion Problem by Newton's Impact Method

The author uses Newton`s impact method to compute the centripetal force exerted on a particle moving uniformly on a circumference of a noneuclidean circle.

##### On the Measure of Solid Angles

The author revisits formulas of measuring solid angles that he could find only in centuries-old literature, and provides modern versions of the proofs.

##### Coin Flipping, Dynamical Systems, and the Fibonacci Numbers

This article links two seemingly unrelated problems, one in probability and the other in dynamical systems, and shows they are actually one involving Fibonacci numbers.

##### Partitions into Consecutive Parts

The authors state and prove a theorem on the number of partitions of an integer into consecutive parts.

##### Proof without Words: Integration by Parts

The rule of integration by parts is proven visually.