# 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.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

Based on the notion of "arithmetic triangles," arithmetic quadrilaterals are defined. It was proved by using an elliptic curve argument that no such quadrilateral can be inscribed on a circle. This capsule provides an elementary proof following the classic ideas of Euler and Fermat.

##### An Improper Application of Green's Theorem

Use Green's Theorem to find a particular integral of a single variable

##### Proof Without Words: Double Angle Formula via Area

A triangle labeled two different ways verifies the double angle formula for sines.

##### Computing the Fundamental Matrix for a Reducible Markov Chain

A Markov chain with 9 states is used to illustrate a technique for finding the fundamental matrix.

##### A Geometric Intrepretation of the Columns of the (Pseudo) Inverse of A

The pseudo inverse of a matrix $$A$$ is used to obtain information about the rows of $$A$$.

##### A Characterization of a Quadratic Function in $$\Re^n$$

The author provides a characterization of $$n$$-dimensional quadratic functions in terms of tangent planes.