## Current Events in Mathematics (5 Need to Knows)

Current events in mathematics point to a growing field with a diversifying range of utilities and practices. The history of mathematics is an interesting dive into the past, present and...

## Confluent Relations (using Reduction Relations)

We discuss confluent relations; in particular, we prove Newman's Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation....

## Well-Founded Relations (and Well-Founded Induction)

Well-Founded induction is a generalization of mathematical induction. Well-Founded Induction Definition. Let $\longrightarrow$ be a relation on $X.$ 1) If $A\subseteq X$ and $a\in A,$ then $a$ is called a...

## Partial Order Relations (Mappings on Ordered Sets)

We discuss many properties of ordered sets including Noetherian ordered sets and order ideals. We also detail monotone mappings and isomorphisms between ordered sets. Ordered Sets Throughout we assume $(X,\geq)$...

## Equivalence Relations (Properties and Closures)

Equivalence Relations We discuss the reflexive, symmetric, and transitive properties and their closures. The relationship between a partition of a set and an equivalence relation on a set is detailed....

## Binary Relations (Types and Properties)

Let $X$ be a set and let $X\times X=\{(a,b): a,b \in X\}.$ A (binary) relation $R$ is a subset of $X\times X$. If $(a,b)\in R$, then we say $a$ is related to $b$...

## Set Theory (Basic Theorems with Many Examples)

We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and Cartesian products. We also demonstrate how to work with families of sets. For a...

## Canonical Forms and Jordan Blocks

We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into linear polynomials. We demonstrate this...

## Green’s Theorem (by Example)

Green’s Theorem for Simply Connected Regions Green's Theorem is named after the mathematician George Green. Theorem. (Green's Theorem) Let $R$ be a simply connected region with a piecewise smooth boundary...

## Probability Density Functions (Applications of Integrals)

Applications of Integrals We will consider the following applications: average value of a function over a region, mass of a lamina, electric charge, moments and center of mass, moments of...

## Invariant Subspaces and Generalized Eigenvectors

Invariant Subspaces of a Linear Transformation We let $V$ and $W$ denote real or complex vector spaces. Suppose $T \in \mathcal{L}(V)$ and $U$ a subspace of $V$, then we say...

## Conservative Vector Fields and Independence of Path

Independence of Path Definition. The line integral is called independent of path if in a region $D$, if for any two points $P$ and $Q$ in $D$ then the line...

## Diagonalization of a Matrix (with Examples)

An $n\times n$ matrix $A$ is called diagonalizable if $A$ is similar to some diagonal matrix $D.$ If the matrix of a linear transformation $T$ with respect to some basis...

## Line Integrals (Theory and Examples)

Definition of a Line Integral Let $C$ be a smooth curve, with parametric equations $x=x(t),$ $y=y(t),$ and $z=z(t)$ for $a\leq t\leq b,$ that lies within the domain of a function...

## Eigenvalues and Eigenvectors (Find and Use Them)

Eigenvalues and Eigenvectors Let $\mathbb{F}$ be either the real numbers or the complex numbers. A nonzero vector $v$ in $\mathbb{F}^n$ is called an eigenvector of an $n\times n$ matrix $A$...

## Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the curl of $\mathbf{V}$ is defined...

## Determinant of a Matrix (Theory and Examples)

Determinants and Trace The determinant function can be defined by essentially two different methods. The advantage of the first definition, one which uses permutations, is that it provides an actual...

## Vector Fields and Gradient Fields

Introduction to Vector Fields Definition. A vector field in $\mathbb{R}^n$ is a function $\mathbf{V}$ that assigns a vector from each point in its domain. A vector field with domain $D$...