Science and Math Articles

Here is the collection of expert STEM articles that span all the sciences and mathematics. Some examples include physics, mathematics, biology, chemistry, cosmology, and earth sciences. For math, we cover algebra to advanced calculus.  These articles are all peer-reviewed by the community.

Inside the Scientific Box: History and Challenges Today

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In Memoriam In Memory of Dr. Thomas J. LeCompte (1964-2025), Detector Designer and Champion of Education and Science. Prologue Defining "the box" Someone…
Entangled Photon Polarization Qubits

Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality per Quantum Information Theory

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In her YouTube video Bell's Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell's inequality. In…
relativator

Relativator (Circular Slide-Rule) – Simulated with Desmos

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The Relativator (revisited) This is an update of my 2006 post (reconstructed in 2014) Relativator: The circular slide-rule for physicists. This is a…
dirac function

What Exactly is Dirac’s Delta Function?

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Introduction: "Convenient Notation"In Dirac's Principles of Quantum Mechanics published in 1930 he introduced a "convenient notation" he referred …
quantum entanglement

Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

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Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and…
complex numbers

Fixing Things Which Can Go Wrong With Complex Numbers

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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

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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

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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…
no tidal bulge

Why the Tidal Bulge Doesn’t Exist – Tidal Theory Explained

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[CONTENT] Introduction Overview That there is no tidal bulge is the key premise of this article. Upper-level oceanography undergraduates and above know…
topology

The Many Faces of Topology

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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

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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,…
physics of reality

Quantum Reconstruction & Bell Correlations Explained

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Comment on PBS Space Time video PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their…
pvsnp

Aspects Behind the Concept of Dimension in Various Fields

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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

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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…

Addition of Velocities (Velocity Composition) in Special Relativity

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The "Addition of Velocities" formula (more correctly, the "Composition of Velocities" formula) in Special Relativity[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c}…
Schrodinger's cat

Schrödinger’s Cat and the Qbit

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The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##\ddot{\text{o}}##dinger noted in 1935 writing [1],…
slinky drop experiment

The Slinky Drop Experiment Analysed

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The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye.…
infinite universe

Spatial Curvature in Cosmological Models

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Standard cosmological models are classified by spatial curvature into three broad types: open, flat, and closed universes. These correspond to negative,…
Lambert W Function in Finance

The Lambert W Function in Finance

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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 a Bad Idea

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

Series in Mathematics: From Zeno to Quantum Theory

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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

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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…
Milli-Ohm

The Poor Man’s Milli-Ohm Meter

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Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article,…
Differential Equation Systems and Nature

Differential Equation Systems and Nature

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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

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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?

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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

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Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…
chatgpt-reliable

Why ChatGPT Is Unreliable: Design Limits and Misconceptions

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ChatGPT's reliability I'll start with the simple fact: ChatGPT is not a reliable answerer to questions. Wolfram's explanation To try to explain why from…
quantum information

How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics

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In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit…
Infinitesimals

What Are Infinitesimals – Advanced Version

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

Popular Science Communication: Bridging Science & Public

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Abstract Famous Science Communicators There is so much to say about the many endeavors by professional scientists to explain to us the world. The list…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

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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

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

Why There Are Maximum Mass Limits for Compact Objects

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In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Implications

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder…
tensors relativity

What Are Tensors and Why Are They Used in Relativity?

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If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline…
gravity collapse

Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational…
twin paradox

Twin Paradox Guide: Spacetime Geometry Explained Clearly

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Introduction This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads…
gravity collapse

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

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Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the…
math classifications

Mathematics Fields & Applications: A Structured Guide

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Introduction I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what…
history of numbers

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

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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

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 * 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

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I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
pvsnp

P vs. NP and what is a Turing Machine (TM)?

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P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and…
quantum computers 101

Quantum Computing Explained: Qubits, Algorithms & Applications

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Overview This article provides an accessible introduction to quantum computing, a cutting-edge technology that processes certain types of information…
gauss law misconceptions

A Physics Misconception with Gauss’ Law

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Introduction It is relatively common to see the following type of argument: The surface area is ##A## and the enclosed charge is ##Q##. The electric…
why can we jump

How Can We Jump When the Ground Does No Work?

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Common mistaken argument It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

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Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…
magnetism current

Symmetry Arguments and the Infinite Wire with a Current

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Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric…
Hyperbola

The Amazing Relationship Between Integration And Euler’s Number

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We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that…
mermin device quantum entanglement

Superdeterminism and the Mermin Device

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Superdeterminism as a way to resolve the mystery of quantum entanglement is generally not taken seriously in the foundations community, as explained in…
relativity rotated graph

Relativity on Rotated Graph Paper (a graphical motivation)

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(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184…
probabilities virus testing

Probabilistic Factors Involved in Disease and Virus Testing

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Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,…
quantum mechanics new approach

Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

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This Insight article presents the main features of a conceptual foundation of quantum physics with the same characteristic features as classical physics…
quantum information

How Quantum Information Theorists Revealed the Relativity Principle at the Foundation of Quantum Mechanics

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This Insight is a condensed version of No Preferred Reference Frame at the Foundation of Quantum Mechanics. Reference numbers here correspond to that paper.…
quantum double slit experiment

Quantum Mechanics and the Famous Double-slit Experiment

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Double-slit Key PointsQuantum mechanics is known for its strangeness, including phenomena like wave–particle duality, which allows particles to…
safety

Safety Factors: Why Forums Avoid Design Advice Online

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Forum Guidelines There are often questions on Physics Forums related to building machines or structures that are left unanswered because of this guideline…
blackboard

10 Math Things We All Learnt Wrong At School

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Some of these could even lead to heavy debates within the scientific community, so maybe I should say: from my point of view. So before you get excited…
tolman law beginner

Tolman Law in a Nutshell

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The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic…
Bayesian statistics part 4

Posterior Predictive Distributions in Bayesian Statistics

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Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticiansRead part 1: How to Get Started with Bayesian StatisticsRead…
bayesian inference

How Bayesian Inference Works in the Context of Science

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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

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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

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

Electric Vector Potential: Detect Non-Conservative Fields

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Main Point The electric vector potential offers a means of determining the non-conservative component of a mixed stationary or quasi-stationary electric…
Learn the Math

I Know the Math Says so, but Is It Really True?

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I'm sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I'll use an example based on threads…
block universe time illusion

Does the Block Universe of Physics Mean Time is an Illusion?

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I have written many Insights (and coauthored an entire book) explaining how the puzzles, problems, and paradoxes of modern physics can be attributed to…
flight simulator

Introduction to Pitot-Static Systems in Gliders

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Introduction Like many others, I have been seeking new and fun things to do during a pandemic.   I decided on the new Microsoft Flight Simulator 2020,…
mechanical conservation of energy

Why Bother Teaching Mechanical Energy Conservation?

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Note: It is assumed that the reader has read part I and part II of the series. Earlier partsIs Mechanical Energy Conservation Free of Ambiguity? …
kerr spacetime centrifugal force

Intro to Centrifugal Force Reversal in Kerr Spacetime

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In this article, we will analyze "centrifugal force reversal" in Kerr spacetime, similar to what was done for Schwarzschild spacetime in a previous Insights…
physics mysteries

A Principle Explanation of the “Mysteries” of Modern Physics

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All undergraduate physics majors are shown how the counterintuitive aspects ("mysteries") of time dilation and length contraction in special relativity…
Compton Scattering

Exploring the Anatomy of Compton Scattering

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Introduction In this article, we take as our starting point the original equations which Compton drew up and solved in his ground-breaking 1925 article:  From…
Twins Paradox

Learning the Twin Paradox for Freely-falling Observers

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The "twin paradox" is often discussed in the introductory treatment of special relativity. Under "twin paradox" we understand the fact that if two twins…
electromagentic waves traverse

Are Electromagnetic Waves Always Transverse? Full Explanation

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In this insight, we will explore classical electrodynamics and examine whether electromagnetic (EM) waves are always transverse. We use Jefimenko's equations…
pressure is a source of gravity

Is Pressure A Source Of Gravity?

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In a previous series of articles, I posed the question "Does Gravity Gravitate?" and explained how, depending on how you interpreted the terms "gravity"…
Compton Scattering

Massive Meets Massless: Compton Scattering Revisited

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Introduction In a previous article entitled "Alternate Approach to 2D Collisions" we analyzed collisions between a moving and stationary object by defining…
Science Authority

How Non-Scientists Should Evaluate Scientific Claims

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How should non-scientists evaluate scientists' public claims? In a previous article, I discussed why scientists (among others) are rarely interested…
dark energy supernova

Dark Energy Part 3: Fitting the SCP Union 2.1 Supernova Data

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In Part 1 of this 3-part series, I explained kinematics in Einstien-deSitter (EdS) cosmology and in Part 2, I explained kinematics in ##\Lambda##CDM cosmology…
dark energy cosmology lcdm

Dark Energy Part 2: LCDM Cosmology

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This is Part 2 of a 3-part series explaining evidence for so-called “dark energy” leading to a current positive cosmological acceleration. The evidence…
beginner guide to baryons

A Beginner Physics Guide to Baryon Particles

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Baryon Introduction At the beginning of the 20th century, it was thought that all matter consisted of only three particles: the electron, the neutron,…
symmetric spacetime

Slowly Lowering an Object in a Static, Spherically Symmetric Spacetime

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In the first two articles in this series, we looked at the Einstein Field Equation and Maxwell's Equations in a static, spherically symmetric spacetime.…
supernova refsdal

Supernova Refsdal: Multiply-Imaged Supernova Guide

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Supernova (SN) Refsdal — Key PointsAstronomers have been trying to understand space for thousands of years. Supernova Refsdal was the first known…
Dark Energy Cosmology Part 1

Dark Energy Part 1: Einstein-deSitter Cosmology

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In this 3-part series, I want to motivate the (re)introduction of the cosmological constant ##\Lambda## into Einstein's equations of general relativity…
Cosmic Web Connections

Cosmic Web Connectivity: How It Shapes Galaxy Evolution

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Introduction The universe was not perfectly uniform at the beginning; some regions had higher density than others. Over time these higher-density regions…
Gaia search for black holes

Gaia Astrometry: Detecting Gravitational Waves Guide

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Overview Gravitational waves (GWs) are disturbances in spacetime produced by massive objects moving asymmetrically. Only the most massive and most relativistic…
uranus spin

Why Uranus Spins on Its Side: Tilt, Theories, Models

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Uranus' Peculiar Tilt Uranus spins on its side. It has an obliquity (tilt) of 98° — its rotation axis lies closer to the ecliptic plane than any other…
black holes are not empty

VHE Gamma Rays from Sgr A*: Black Holes and CTA Observations

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Black Hole Key PointsBlack holes (BHs) have a gravitational pull so strong that nothing, not even light, can escape. BHs can be divided into two…
energy conservation can we do better

Can We Do Better Than Mechanical Energy Conservation?

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Note: It is assumed that the reader has read part I of the series. Introduction The ambiguity and flaws discussed in part I can be resolved using the…
energy conservation ambiguity

Is Mechanical Energy Conservation Free of Ambiguity?

Introduction "Close to any question that is in the textbook, there is another question that has never been answered that is interesting." [Stephen Wolfram,…
What is Evolution

Introduction to Evolution: Key Concepts & FAQs Guide

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Introduction to Evolution Overview The study of evolution has dominated biology for over a century. Explaining why life is the way it is fascinates both…
mary Somerville

Mary Somerville: Scottish Polymath & RAS Honorary Member

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Mary Somerville & Caroline Herschel: Google DoodleGoogle Doodle: Mary Somerville and Caroline Herschel The Google Doodle for 2 February 2020 celebrated…
fluid dynamics experiment

Video Analysis of Spheres Falling Through Two Liquids

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Introduction Providing accurate fluid-dynamics experiments for undergraduate laboratories is challenging in several ways, including reproducibility, simplicity,…
how to identify wood

Tell One Wood from Another: Basic Wood Anatomy

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Introduction About the author As a long-time woodworker, I enjoy being able to tell one wood from another. In the process of learning how to do that,…
Diffraction Grating Spectrometer.

Learn the Fundamentals of the Diffraction Grating Spectrometer

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Introduction In this article, we will discuss the fundamentals of the diffraction grating spectrometer.  The operation of the instrument is based upon…
hydrogen

Intro to the Ionization Energy of Atomic Hydrogen

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Introduction In previous articles relating to various transition energies in Hydrogen, Helium, and Deuterium we have employed the following formula for…
quantum entanglement

Answering Mermin’s Challenge with the Relativity Principle

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Note: This Insight was previously titled, "Answering Mermin's Challenge with Wilczek's Challenge." While that version of this Insight did not involve any…
Conservation of Spin Angular Momentum

Exploring Bell States and Conservation of Spin Angular Momentum

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In a recent thread, I outlined how to compute the correlation function for the Bell basis states\begin{equation}\begin{split}|\psi_-\rangle &=…
Kuhnian Revolution

Modern Physics Understood as an Unrecognized Kuhnian Revolution

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People often claim on Physics Forums and in the foundations community proper that quantum mechanics is "incomplete." Indeed, Lee Smolin recently wrote…
DC straight wire

Relativistic Treatment of the DC Conducting Straight Wire

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Introduction The direct-current-conducting infinitely long wire is often discussed in the context of relativistic electrodynamics. It is of course a completely…
Deuterium Lyman Alpha Line

Revisiting The Deuterium Lyman Alpha Line Experiment

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Introduction In this article, we will be revisiting a somewhat understudied (and seemingly unrepeated) experiment to measure the Deuterium Lyman Alpha…
Solar Balmer Lines

Signature of the Sun: Analyzing the Solar Balmer Series Lines

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Introduction We are in many ways a fortunate generation having so many wonderful tools at our disposal on account of the "silicon revolution". The advent…