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.

Lambert W Function in Finance

The Lambert W Function in Finance

/
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

/
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

/
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

/
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

/
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

/
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

/
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?

/
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

/
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 Not Reliable

/
I'll start with the simple fact: ChatGPT is not a reliable answerer of questions.To try to explain why from scratch would be a heavy lift, but fortunately,…
quantum information

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

/
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

/
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

Opinion: When Pro Scientists Explain Using Pop Science

/
Abstract There is so much to say about the many endeavors by professional scientists to explain to us the world. The list is long: Carl Sagan, Harald…
integration and complex differentiation

An Overview of Complex Differentiation and Integration

/
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

/
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

/
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

/
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?

/
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

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

When Discussing the Twin Paradox: Read This First

/
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 here on…
gravity collapse

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

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

Classification of Mathematics by 42 Branches

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

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

/
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

/
 * 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

/
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)?

/
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 for Beginners

/
Introduction to Quantum Computing This introduction to quantum computing is intended for everyone and especially those who have no knowledge of this relatively…
gauss law misconceptions

A Physics Misconception with Gauss’ Law

/
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?

/
It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the ground, the ground does…
Riemann Hypothesis History

The History and Importance of the Riemann Hypothesis

/
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

/
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

/
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

/
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)

/
(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

/
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

/
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

/
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

/
Key PointsQuantum mechanics is known for its strangeness, including phenomena like wave-particle duality, which allows particles to behave like waves. …
safety

When Science is Not Enough: Legal and Health Implications of Questions

/
There are often questions on Physics Forums related to building machines or structures that are left unanswered because of this guideline (found in the…
blackboard

10 Math Things We All Learnt Wrong At School

/
The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot…
tolman law beginner

Tolman Law in a Nutshell

/
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

/
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

/
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

/
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

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

Introduction to Electric Vector Potential and Its Applications

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

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

/
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?

/
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

/
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, but the Condor…
mechanical conservation of energy

Why Bother Teaching Mechanical Energy Conservation?

/
Note: It is assumed that the reader has read part I and part II of the series.Is Mechanical Energy Conservation Free of Ambiguity? Can We Do Better…
kerr spacetime centrifugal force

Intro to Centrifugal Force Reversal in Kerr Spacetime

/
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

/
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

/
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

/
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?

/
In this insight, we shall endeavor in the realm of classical electrodynamics and examine whether EM waves are always transverse. We shall make use of Jefimenko's…
pressure is a source of gravity

Is Pressure A Source Of Gravity?

/
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

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

Is Science an Authority? How to View Announcements from Scientists

/
In a previous article, I discussed why scientists (among others) are almost never interested in considering a new theory proposed by a non-scientist: because…
dark energy supernova

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

/
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

/
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

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

Slowly Lowering an Object in a Static, Spherically Symmetric Spacetime

/
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

The Reappearance of Supernova (SN) Refsdal

/
Key Points Astronomers have been trying to understand space for thousands of years. Supernova Refsdal was the first known multiply imaged supernova. …
Dark Energy Cosmology Part 1

Dark Energy Part 1: Einstein-deSitter Cosmology

/
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

The Evolution of the Universe, Cosmic Web and Connections

/
The universe was not perfectly uniform when it started, some areas had a higher density than others. During the evolution of the universe, these areas…
Gaia search for black holes

Gaia and the Race to Detect More Gravitational Waves

/
Gravitational waves (GW’s) are disturbances in spacetime produced by any massive object moving asymmetrically. However, only the most massive and most…
uranus spin

3 Possible Models For Why Uranus Spins on Its Side

/
Uranus spins on its side. Uranus has an obliquity (tilt) of 98º, making its axis of rotation closer to the ecliptic plane than any other planet. It is…
black holes are not empty

Black Holes Are Not Empty Voids. A Black Hole at the Heart!

/
Key PointsBlack Holes (BH’s) have a gravitational pull so strong that nothing, not even light, can escape it BH’s can be divided into two…
energy conservation can we do better

Can We Do Better Than Mechanical Energy Conservation?

/
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

What is Evolution: A Beginner’s Guide

/
Introduction to Evolution The study of evolution has dominated the field of biology for over a century. Explaining why the life we see is as it is fascinating…
mary Somerville

Mathematician Mary Somerville Features in Google Doodle

/
The Google Doodle for 2 February 2020 celebrated Mary Somerville, the Scottish polymath and science writer, and with Caroline Herschel, the joint first-ever…
fluid dynamics experiment

Accurate Fluid Dynamics By Video Analysis

/
IntroductionProviding accurate fluid dynamics experiments for undergraduate laboratories is challenging in several ways, including reproducibility,…
how to identify wood

Want to Tell One Wood from Another? Learn Basic Wood Anatomy

/
As a long-time woodworker, I have an interest in being able to tell one wood from another and in the process of learning how to do that, I've developed…
Diffraction Grating Spectrometer.

Learn the Fundamentals of the Diffraction Grating Spectrometer

/
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

/
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

/
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

/
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

/
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

/
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

/
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

/
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…
Bohr’s Helium Lines

Understanding Bohr’s Helium Lines

/
Introduction In a previous article "Calculating the Balmer Alpha Line" we mentioned how accurate predictions of the spectral lines of singly ionized Helium…
rotating room

How to Zip Through a Rotating Tunnel Without Bumping Into the Walls

/
Preface While browsing through unanswered posts in the Classical Physics Workshop, I came across a gem at the link shown below.  For the reader's convenience,…
Balmer Alpha Line

Calculating the Balmer Alpha Line: Atomic Hydrogen

/
Introduction Most readers acquainted with the hydrogen spectrum will be familiar with the set of lines in the visible spectrum representing transitions…
spacetime maxwell

Maxwell’s Equations in a Static, Spherically Symmetric Spacetime

/
In the first article in this series, we looked at the Einstein Field Equations in a static, spherically symmetric spacetime. In this article, we are going…
quantum superposition

Understanding Superposition Physically and Mathematically

/
What is superposition Superposition is a fundamental concept in physics and mathematics, particularly in the field of wave theory and quantum mechanics.…
What is Entropy

What Thermodynamics and Entropy Means

/
Introduction The student of thermodynamics, as they consider pistons and ideal gasses and such, often begin to grasp the nature of entropy only to find…
Einstein Field Equation

The Einstein Field Equation in a Static, Spherically Symmetric Spacetime

/
This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmetric spacetimes. This is a…
writing proofs

How to Write a Math Proof and Their Structure

/
Proofs in mathematics are what mathematics is all about. They are subject to entire books, created entire theories like Fermat's last theorem, are hard…
Star Equations

Equations of State for Photon Gas and Relativistic Electron Gas

/
This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a…
rules of quantum mechanics

The 7 Basic Rules of Quantum Mechanics

/
For reference purposes and to help focus discussions on Physics Forums in interpretation questions on the real issues, there is a need for fixing the common…
Quantum Particles

Clarifying Common Issues with Indistinguishable Particles

/
Commonly there is a lot of imprecision in talking about ''indistinguishable'' (or ''identical'') particles, even in serious work. This Insight article…
Renewable Energy Power Grid

When Renewable Energy Meets Power Grid Operations

/
The electric power industry faces much turmoil in the coming decades.  The business model of the electric utility company (public or private) may not…
Maupertuis Principle

A Pure Hamiltonian Proof of the Maupertuis Principle

/
Here is another version of proof of Maupertuis's principle. This version is pure Hamiltonian and independent of the Lagrangian approach.The proof…
create a universe story

How to Create a Universe – Instructions for an Apprentice God

/
(-: ##~~~## A fantasy to be read at leisure time ##~~~## :-) Written October 7, 1999 to relax from the strains of working on a new interpretation of quantum…
quantum qubit

A Classical View of the Qubit

/
This Insight article is part of my paper Foundations of quantum physics III. Measurement, featuring the thermal interpretation of quantum physics. See…
The Sum of Geometric Series from Probability Theory

The Sum of Geometric Series from Probability Theory

/
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…
quantum mechanics is incomplete

9 Reasons Quantum Mechanics is Incomplete

/
I argue that all interpretations of quantum mechanics (QM) are incomplete, each for its own reason. I also point out that for some interpretations (those…
against quantum interpretations

Against “interpretation”

/
I am against "interpretations" of Quantum Mechanics (QM) in a sense in which John Bell [1] was against measurement in QM and Travis Norsen [2] is against…