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.

no tidal bulge

Beyond the Tidal Bulge

/
8 Comments
Overview That there is no tidal bulge is the key premise of this article. Upper-level oceanography undergraduates and above know this. Yet the tidal bulge…
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,…
physics of reality

PBS Video Comment: “What If Physics IS NOT Describing Reality”

/
3 Comments
PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their video What If Physics IS NOT Describing…
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…

Addition of Velocities (Velocity Composition) in Special Relativity

/
2 Comments
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

/
27 Comments
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

/
1 Comment
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

Is the Universe Finite, or Is It Infinite?

/
0 Comments
Standard cosmological models come in three flavors, open, flat, and closed,[Carroll] whose spatial curvatures are negative, zero, and positive. The open…
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 a Bad Idea

/
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 allowed? Then…
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…
Milli-Ohm

The Poor Man’s Milli-Ohm Meter

/
0 Comments
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

/
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…
chatgpt-reliable

Why ChatGPT AI Is Not Reliable

/
138 Comments
I'll start with the simple fact: ChatGPT is not a reliable answerer to 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

/
0 Comments
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

/
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…
Carl Sagan

When Pro Scientists Explain Using Pop Science

/
19 Comments
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

/
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…
white dwarfs

Why There Are Maximum Mass Limits for Compact Objects

/
0 Comments
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

/
0 Comments
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?

/
0 Comments
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

/
0 Comments
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

/
0 Comments
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

/
0 Comments
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

/
0 Comments
 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

/
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,…
pvsnp

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

/
1 Comment
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

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

A Physics Misconception with Gauss’ Law

/
3 Comments
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?

/
20 Comments
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

/
0 Comments
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

/
0 Comments
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

/
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…
mermin device quantum entanglement

Superdeterminism and the Mermin Device

/
140 Comments
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)

/
0 Comments
(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

/
2 Comments
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

/
176 Comments
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

/
12 Comments
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

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

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

/
7 Comments
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

/
10 Comments
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

/
46 Comments
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

/
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…
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…
electric potential

Introduction to Electric Vector Potential and Its Applications

/
16 Comments
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?

/
104 Comments
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?

/
29 Comments
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

/
1 Comment
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?

/
7 Comments
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

/
0 Comments
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

/
96 Comments
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

/
1 Comment
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

/
1 Comment
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?

/
5 Comments
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?

/
11 Comments
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

/
0 Comments
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

/
66 Comments
In a previous article, I discussed why scientists (among others) are rarely 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

/
0 Comments
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

/
0 Comments
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

/
9 Comments
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

/
0 Comments
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

/
0 Comments
Supernova (SN) Refsdal Key Points Astronomers 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

/
6 Comments
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

/
1 Comment
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

/
1 Comment
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

/
4 Comments
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

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

Can We Do Better Than Mechanical Energy Conservation?

/
6 Comments
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

/
1 Comment
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

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

Accurate Fluid Dynamics By Video Analysis

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

Tell One Wood from Another: Basic Wood Anatomy

/
12 Comments
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

/
11 Comments
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

/
2 Comments
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

/
89 Comments
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

/
0 Comments
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

/
16 Comments
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

/
11 Comments
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

/
0 Comments
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

/
2 Comments
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

/
0 Comments
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

/
2 Comments
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

/
0 Comments
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

/
11 Comments
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

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

What Thermodynamics and Entropy Means

/
1 Comment
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

/
15 Comments
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

/
1 Comment
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…