# Mathematics Articles

## Mathematics (the Ultimate Guide to Becoming a Mathematician)

Mathematics Articles Surprisingly, mathematics has no widely accepted definition. It has been defined in various ways for centuries, such as the science of quantity, logic, or intuition. A few main schools of thought exist today about how to formally define it, but let’s leave that for more in-depth mathematics articles. In a nutshell, you can …

## Current Events in Mathematics (5 Stories You Need to Know)

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 future. It’s long been one of the most studied disciplines. In fact, it’s older than most other fields of study in general. To be fair, …

## 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. We also give a generalization of Newman’s lemma based on the Buchberger-Winkler’s Property. Reduction Relations Let $\longrightarrow$ be a relation on $X.$ If there exists  …

## 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 $\longrightarrow$-minimal element of $A$ if  there does not exist $b\in A$ such that $a\longrightarrow b.$ 2) If each nonempty subset of $X$ has a $\longrightarrow$-minimal …

## 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 $(X,\geq)$ is an ordered set.  By this we mean that $X$ is a set and that $\geq$ is binary relation on $X$ that is reflexive, antisymmetric, …

## 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. We then give the two most important examples of equivalence relations. Reflexive, Symmetric, and Transitive Relations Definition. Let $X$ be a set. A relation $R$ …

## 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 $b$ by $R$. It is possible to have both $(a,b)\in R$ and $(a,b’)\in R$ where $b’\neq b$; that is any element in $X$ could be related to …

## Composition of Functions and Inverse Functions

Composition Theorem. Let $f:X\to Y$ be a function. If $g:Y\to Z$ and $g\circ f$ is injective, then $f$ is injective. Proof. Let $x_1, x_2\in X$. Then $$f(x_1)=f(x_2) \Longrightarrow (g\circ f)(x_1)=(g\circ f)(x_2) \Longrightarrow x_1=x_2$$ shows that $f$ is injective. Theorem. Let $f:X\to Y$ be a function.  If $g:Y\to Z$ and $g\circ f$ is surjective, …

## One-to-One Functions and Onto Functions

Injective Functions Definition. Let $X$ and $Y$ be sets. A function $f:X\to Y$ is called injective if  $$\forall x_1,x_2\in X, f(x_1)=f(x_2)\implies x_1=x_2.$$ Theorem. Let $f:X\to Y$ be a function.  If $f$ is injective and $A\subseteq X$, then $f|_A$ is injective. Proof. Let $x_1, x_2\in A$.  Then  $$f|_A(x_1)=f|_A(x_2) \Longrightarrow f(x_1)=f(x_2) \Longrightarrow x_1=x_2$$ …

## Functions (Their Properties and Importance)

Let $X$ and $Y$ be sets, we say $f$ is a function from $X$ to $Y$ if $f$ is a subset of $X\times Y$ such that the domain of $f$ is $X$ and $f$ has the property:  if $(x,y)\in f$ and $(x,z)\in f$ then $x=z$. For each $x\in X$, the unique $y\in Y$ such that …

## Families of Sets (Finite and Arbitrarily Indexed)

Families of Sets Finite Unions and Intersections Given any $n$ sets $A_1, A_2, \ldots, A_n$, we define their union to be the set $$A_1 \cup A_2 \cup \cdots \cup A_n =\{x\mid x\in A_i \text{ for some } i, 1\leq i \leq n\}.$$ and their intersection to the the set   A_1 \cap A_2 \cap \cdots …

## 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 brief discussion of the reviews of (elementary) Halmos’ Naive Set Theory read this. A solid understanding of propositional and predicate logic is strongly recommended. To …

## 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 with an example and provide several exercises. Jordan Basis A basis of $V$ is called a Jordan basis for $T$ if with respect to 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 curve $C$ oriented counterclockwise and let ${F}= M {i}+ N {j}+0 {k}$ be a continuously differentiable vector field on $R,$ then \begin{align*} \oint_C M dx+Ndy= …

## 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 inertia, and probability density functions. Average Value Recall the average value of an integrable function of one variable on a closed interval is the integral …

## 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 $U$ is an invariant subspace of $T$ if $u\in U$ implies $Tu\in U.$ Lemma. Suppose $T \in \mathcal{L}(V)$ and $U$ a subspace of $V.$ Then …

## 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 integral along every piecewise smooth curve in $D$ from $P$ to $Q$ has the same value. Theorem. (Independence of Path) If $V$ is a continuous …

## 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 is diagonal then we call $T$ diagonalizable. Diagonalization Theorem Theorem. An $n\times n$ matrix $A$ is diagonalizable if and only if it has $n$ linearly …

## 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 $f(x,y,z).$ We say that $C$ is orientable if it is possible to describe direction along the curve for increasing $t.$ Partition $C$ into $n$ sub-arcs, …

## 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 $A$ if $A v$ is a scalar multiple of $v$, that is $A v= \lambda v$ for some scalar $\lambda.$ Note that this scalar $\lambda$ may …

## 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 by curl $\mathbf{V}=\nabla \times \mathbf{V}$ where $$\nabla =\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}k$$ is the del operator. For a discussion on the physical …

## 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 formula for $\det(A)$, a fact of theoretical importance. The disadvantage is that, quite frankly, computing a determinant by this method can be cumbersome. A pattern …

## 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 $D$ in $\mathbb{R}^n$ has the form $${V}(x_1, \ldots, x_n) = \langle u_1 (x_1, \ldots,x_n), \ldots, u_n (x_1, \ldots, x_n) \rangle$$ where the scalar functions …

## Inner Products and Orthonormal Bases

Recall that the norm of $x\in \mathbb{R}^n$ defined by $\left|\left |x\right|\right | = \sqrt{x_1^2+x_2^2}$ is not linear. To injective linearity into the discussion we introduce the dot product: for $x,y\in \mathbb{R}^n$ the dot product of $x$ and $y$ is defined as $x \cdot y=x_1 y_1+\cdots +x_n y_n.$ Obviously $x \cdot x=\left|\left |x\right|\right |^2$, and with …