Mathematics

ai problem solving

AI Enriched Problem Solving

/
1 Comment
Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems…
dirac function

What Exactly is Dirac’s Delta Function?

/
0 Comments
Introduction: "Convenient Notation"In Dirac's Principles of Quantum Mechanics published in 1930 he introduced a "convenient notation" he referred …
complex numbers

Fixing Things Which Can Go Wrong With Complex Numbers

/
0 Comments
Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2…
Fermat's Last Theorem

Fermat’s Last Theorem

/
1 Comment
Abstract Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states…
vector spaces

Vector Spaces: Concepts, History, and Applications Guide

/
1 Comment
The Concept A vector space is an additively written abelian group together with a field that operates on it.Intuitive picture vs abstract definition Vector…

Groups, The Path from a Simple Concept to Mysterious Results

/
9 Comments
Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement…
topology

The Many Faces of Topology

/
0 Comments
Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see…
brownian motion

Brownian Motions and Quantifying Randomness in Physical Systems

/
1 Comment
Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It's now a key player in data science,…
pvsnp

Aspects Behind the Concept of Dimension in Various Fields

/
2 Comments
Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,…
complex numbers views

Views On Complex Numbers

/
70 Comments
Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted…
Lambert W Function in Finance

The Lambert W Function in Finance

/
0 Comments
Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…
infinity

Why Division by Zero Is Impossible: 10 Mathematical Reasons

/
33 Comments
A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was…
geometric series

Series in Mathematics: From Zeno to Quantum Theory

/
2 Comments
Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…
Epsilontic limits and continuity

Epsilontic – Limits and Continuity

/
1 Comment
Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I…
Differential Equation Systems and Nature

Differential Equation Systems and Nature

/
12 Comments
Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural…
calc precalc

Beginners Guide to Precalculus, Calculus and Infinitesimals

/
3 Comments
Introduction I am convinced students learn Calculus far too late.   In my view, there has never been a good reason for this.In the US, they go through…
what are numbers

What Are Numbers?

/
7 Comments
Introduction When doing mathematics,  we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive.   Going…
world of algebras

Introduction to the World of Algebras

/
0 Comments
Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…
Infinitesimals

What Are Infinitesimals – Simple Version

/
5 Comments
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
Infinitesimals

What Are Infinitesimals – Advanced Version

/
0 Comments
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…
art of integration

The Art of Integration

/
0 Comments
Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

/
0 Comments
Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments…
lie group physics

When Lie Groups Became Physics

/
0 Comments
Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
recursion

Reduction of Order For Recursions

/
0 Comments
This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…
history of numbers

Counting to p-adic Calculus: All Number Systems That We Have

/
0 Comments
An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim…
evariste galois

Évariste Galois and His Theory

/
0 Comments
 * Oct. 25th, 1811  † May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave…
definition differences

Yardsticks to Metric Tensor Fields

/
1 Comment
I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

/
1 Comment
Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
hypercube integral

A Novel Technique of Calculating Unit Hypercube Integrals

/
1 Comment
Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first…
Riemann Hypothesis

The Extended Riemann Hypothesis and Ramanujan’s Sum

/
2 Comments
Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All…
Hyperbola

The Amazing Relationship Between Integration And Euler’s Number

/
0 Comments
We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…
probabilities virus testing

Probabilistic Factors Involved in Disease and Virus Testing

/
2 Comments
Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,…
trig special functions

A Trick to Memorizing Trig Special Angle Values Table

/
3 Comments
In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we…
Theorema Primum tutorial

An Introduction to Theorema Primum

/
0 Comments
Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…
tan rule

Physical Applications of the “Tan Rule”

/
0 Comments
Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,…
euler sums

Investigating Some Euler Sums

/
0 Comments
So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that…
Fourier Series Riemann Zeta Function

Computing the Riemann Zeta Function Using Fourier Series

/
5 Comments
Euler's amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous…
valentines reflections graphs

Valentine’s Reflections: Mathematical Matters of the Heart

/
4 Comments
Introduction Being a somewhat geeky Maths 'nerd', I spent days leading up to Valentine's Day trying to find a Maths function appropriate to the day. In…
Bayesian statistics part 4

Posterior Predictive Distributions in Bayesian Statistics

/
7 Comments
Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticiansRead part 1: How to Get Started with Bayesian StatisticsRead…
impossible triangles

Ambiguous SSA Case in Triangles — Law of Sines Explained

/
0 Comments
What is the ambiguous case? In U.S. high schools, students are often introduced to geometric proofs for the first time. A common task is to use a small…
bayesian inference

How Bayesian Inference Works in the Context of Science

/
12 Comments
Confessions of a moderate Bayesian part 3 Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian ProbabilityBayesian…
bayesian statistics part 2

Exploring Frequentist Probability vs Bayesian Probability

/
38 Comments
Confessions of a moderate Bayesian, part 2 Read Part 1: Confessions of a moderate Bayesian, part 1Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13280237 Background One…
bayesian statistics

How to Get Started with Bayesian Statistics

/
13 Comments
Confessions of a moderate Bayesian, part 1 Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13265286 Background I…
calculus

How to Solve Second-Order Partial Derivatives

/
1 Comment
Introduction A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are…
lerch and zeta functions

The Analytic Continuation of the Lerch and the Zeta Functions

/
0 Comments
Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…
Integral Representations of Some Special Functions

The Orin Fractional Calculus

/
8 Comments
Introduction This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets $$C_N^n:…
Odd Sums

Explore the Fascinating Sums of Odd Powers of 1/n

/
3 Comments
The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is…
SOHCAHTOA

SOHCAHTOA: Seemingly Simple, Conceivably Complex

/
7 Comments
What is SOHCAHTOA SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…
How to Find Potential Functions

How to Find a Potential Function of a Vector Field

/
0 Comments
Definition and summary Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it? Conditions and equations $$\nabla \phi(x,y,z)…
What is a linear equation

First-Order Linear Equation: Definition & Solutions

/
0 Comments
Definition / Summary This article summarizes the first-order (linear) polynomial equation in one variable, its solution, and natural extensions to matrices…
What is a fibre bundle

Fibre Bundle: Definition, Examples & Intuitive Guide

/
3 Comments
Definition / Summary Fibre bundle — intuitively, a fibre bundle is a space E that locally looks like a product B × F but may have a different global…
What are real numbers

Real Numbers: Definition, Properties & Completeness

/
0 Comments
Definition of real numbers Real numbers are the set of all values that can appear on the continuous number line. They include rational numbers (fractions…
integration by parts

What is Integration By Parts? A 5 Minute Introduction

/
0 Comments
Definition/Summary In this article, we shall learn a method for integrating the product of two functions. This method is derived from the 'product rule'…
What is a parabola

Parabola Explained: Properties, Equations & Practical Uses

/
0 Comments
What is a Parabola? A parabola is a U-shaped curve that appears frequently in mathematics, physics and engineering. It is a conic section defined by a…
Limit of a Function

Limits of Functions for Calculus: Definition & Examples

/
0 Comments
What is a limit? In mathematics, a limit describes the behavior of a function or sequence as its input approaches a particular value. Limits are a cornerstone…
surface integral

Demystifying Parameterization and Surface Integrals

/
3 Comments
Introduction This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration…
Lie Algebra Intro

Lie Algebra Basics: Definitions, Equations & Examples

/
0 Comments
Definition / Summary A Lie algebra (pronounced "Lee") is the tangent-space algebra of a Lie group at its identity element. Concretely, it is a vector…
What is a Tangent Line

Tangent Line and Hyperplane: Definitions & Circle Example

/
0 Comments
Definition / Summary Informal idea The tangent to a curve in a plane at a particular point has the same gradient as the curve at that point. Generalization More…
The Sum of Geometric Series from Probability Theory

The Sum of Geometric Series from Probability Theory

/
12 Comments
Here I present a simple (but to the best of my knowledge, new) derivation of the formula for the sum of the infinite geometric series. The derivation is…
lie algebra representations

Learn Lie Algebras: A Walkthrough – The Representations

/
1 Comment
  Part III: Representations  Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…
Lie Algebra Structure

Learn Lie Algebras: A Walkthrough – The Structures

/
2 Comments
  Part II: StructuresDecompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which…
lie algebra basics

Learn Lie Algebras: A Walkthrough – The Basics

/
7 Comments
  Part I: Basics Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…
Calculus Therom

A Numerical Insight for the Fundamental Theorem of Calculus

/
9 Comments
The purpose of this article is not to provide some rigorous statement, neither a rigorous clever proof of the fundamental theorem of calculus. It is rather…
stock options math

Intuitive Black‑Scholes Options Pricing Explained Simply

/
19 Comments
Introduction Financial options — the right to purchase (call) or sell (put) stock or other assets at a fixed price on a future date — have been around…
tensor calculus

The 10 Commandments of Index Expressions and Tensor Calculus

/
16 Comments
Having more than 10 years of experience in teaching vector and tensor calculus and special and general relativity, I have noted that many people have the…
hilbertspaces2

Learn the Basics of Hilbert Spaces and Their Relatives: Operators

/
4 Comments
  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##,…
hilbertspaces

Learn the Basics of Hilbert Spaces and Their Relatives: Definitions

/
2 Comments
  Basics Language first: There is no such thing as the Hilbert space.Hilbert spaces can look rather different, and which one is used in…

Demystifying the Chain Rule in Calculus

/
7 Comments
Introduction There are a  number of posts on PF involving a general confusion over the multi-variable chain rule.  The problem is often caused by…
dice

Learn About Intransitive Dice with a Twist

/
6 Comments
Intransitive dice are sets of dice that don't follow the usual rules for "is better/larger than". If A<B and B<C, then A<C. If Bob runs faster…
qtf_geometry

Learn the Geometry of Mathematical Quantum Field Theory

/
33 Comments
This is the first chapter in a series on Mathematical Quantum Field Theory.The next chapter is 2. Spacetime. 1. Geometry The geometry of physics…
manifold2

A Journey to The Manifold SU(2): Representations

/
3 Comments
Part 1 Image source: [23]  Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

/
3 Comments
Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…
fourierseries2

Learn Further Sums Found Through Fourier Series

/
5 Comments
In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be…
mathoperators

How to Tell Operations, Operators, Functionals, and Representations Apart

/
24 Comments
 All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing.…
Parsevalstheorem

Learn an Integral Result from Parseval’s Theorem

/
5 Comments
Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…
whatisatensor

Tensors Explained: Scalars, Vectors, Matrices & Math

/
58 Comments
Introduction Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can…
deriitive5

The Pantheon of Derivatives – Important Theorems (V)

/
2 Comments
Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||<…
deriitive4

The Pantheon of Derivatives – Lie Derivatives And Others (IV)

/
6 Comments
  Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications…
deriitive3

The Pantheon of Derivatives – Sections, Pullbacks And Pushforwards (III)

/
3 Comments
  Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance,…
deriitive2

The Pantheon of Derivatives – Manifolds And Vector Fields (II)

/
8 Comments
  Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section on complex functions (The Pantheon of Derivatives - Part…
deriitive

The Pantheon of Derivatives – The Direction (I)

/
3 Comments
  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…
integraltrick

Trick to Solving Integrals Involving Tangent and Secant

/
6 Comments
This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So…
cornuspiral

Learn Frenet Equations in 2-D Which Result in the Cornu Spiral

/
3 Comments
This Insights article is intended as an introduction to the Frenet-Serret equations by showing an interesting application that results from a case in two-dimensional…
geometrysimple

When Simple Geometry Unveils Deep Math

/
27 Comments
Introduction It is a remarkable fact that consideration of very elementary concepts in geometry often leads quickly into deep and unexpected mathematical…
FourierSeries

Using the Fourier Series To Find Some Interesting Sums

/
4 Comments
Preliminaries If f(x) is periodic with period 2p and f’(x) exists and is finite for -π<x<π, then f can be written as a Fourier series: [itex]f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}…
representations

Linear Representations and Why Precision is Important in Math

/
2 Comments
First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space…
partial-differentiation

Learn Partial Differentiation Without Tears

/
6 Comments
Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's…
cubicfunction

Learn How to Solve the Cubic Equation for Dummies

/
5 Comments
Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0[/itex], even though you don't really need such a formula,…
omissionguage

Why the Gauge (Henstock-Kurzweil) Integral Matters

/
31 Comments
The gauge integral (Henstock–Kurzweil) The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are…
integralmisconceptions

Integration Paradoxes: Indefinite Integrals and Constants

/
40 Comments
Introduction Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration…
groupsandgeometry

Klein’s Erlangen Program: Groups Define Geometry

/
7 Comments
Klein's Erlangen program: groups and geometry There is a very deep link between group theory and geometry. Sadly, this link is not emphasized in most…
selfstudy

Self-Study Roadmap: Abstract Algebra, Groups to Galois

/
4 Comments
Overview Roadmap There are three major areas of mathematics: geometry, analysis, and algebra. This insight gives a roadmap for learning basic abstract…
computers

Ramsey Theory: Foundations, Generalizations, Key Results

/
5 Comments
Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these six will either be meeting each other for the first…
irrationalnumbers

Mathematical Irrationality for Dummies

/
13 Comments
On one of my restless wanderings around the Internet, I came upon a collection of proofs of the statement that the square root of 2 is an irrational…
logicp3

Scientific Inference: Balancing Predictive Success with Falsifiability

/
9 Comments
  Bayes' Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning.…
selfstudyanalysis

Advanced Real Analysis: Measure & Functional Study Plan

/
14 Comments
Prerequisites If you wish to follow this guide, you should be familiar with analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight for…
linearalgebra_selfstudy

Linear Algebra Roadmap: Prereqs, Books & Topics Guide

/
11 Comments
Overview This article gives a roadmap for students to learn the basics of linear algebra. Aside from calculus, linear algebra is one of the most applicable…
encryption

Is It Possible to Design an Unbreakable Cipher?

/
10 Comments
Is it possible to design an unbreakable cipher? Do methods of encryption exist that guarantee privacy from even the most capable and highly-resourced…
cipher

The Monographic Substitution Cipher: From Julius Caesar to the KGB

/
7 Comments
A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…
geometry1

Pure Geometry Study Guide: Books & Roadmap for Students

/
3 Comments
Introduction to Pure Geometry Why study pure geometry? Geometry is one of the oldest parts of mathematics. It has been studied and advanced by the greatest…