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no tidal bulge

Beyond the Tidal Bulge

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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 is still used to portray why the Moon is receding the Earth. If there is no tidal bulge, some other explanation is in order. That other explanation uses gravitation as…

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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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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”

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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

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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

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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

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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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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

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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

Is the Universe Finite, or Is It Infinite?

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Standard cosmological models come in three flavors, open, flat, and closed,[Carroll] whose spatial curvatures are negative, zero, and positive. The open…
einstein field equations

All About the Einstein Field Equations

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Definition/Summary The Einstein Field Equations (EFE) are a set of ten interrelated differential equations that form the core of Einstein's general theory…
Multi-Atwood Machine Assembly

How to Solve a Multi-Atwood Machine Assembly

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IntroductionThe figure on the right shows a "double-double" Atwood machine with three ideal pulleys and four masses.  All pulleys are released from…
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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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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…
digital audio guide

Digital Filtering and Exact Reconstruction of Digital Audio

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Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function The first link in my insights article has…
digital audio intro

Introduction to Modern Digital Audio Concepts

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IntroductionFirst, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this…
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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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

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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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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…
Variable Mass Systems

How to Apply Newton’s Second Law to Variable Mass Systems

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Introduction The applicability of Newton's second law in the oft-quoted "general form"  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$…
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 AI Is Not Reliable

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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,…
Infinitesimals

What Are Infinitesimals – Simple 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…
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

When Pro Scientists Explain Using Pop Science

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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…
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The Art of Integration

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Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…
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A Lesson In Teaching Physics: You Can’t Give It Away

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A central principle of Physics Forums regarding homework help is not to provide solutions on demand but to guide students along a path to the answer.  The…
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…
Measure Internal Resistance of Battery

How to Measure Internal Resistance of a Battery

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Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery - typically an AA or D…
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…
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When Discussing the Twin Paradox: Read This First

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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

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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…
object slide down ramp physics

Subtleties Overlooked in Friction Questions: Object Slides Down Ramp

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Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ' on the ramp,…
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Classification of Mathematics by 42 Branches

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

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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…
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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…
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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…
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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,…
ATmega8A

Programming an ATmega8A using Arduino

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2 Comments
If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino projects permanent,…
pvsnp

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

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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…
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Quantum Computing for Beginners

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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

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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…
model magnet

How to Model a Magnet Falling Through a Conducting Pipe

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7 Comments
Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features…
model magnet

How to Model a Magnet Falling Through a Solenoid

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11 Comments
Introduction Modeling a magnet realistically is a task best done numerically.  Even the simplified model of two separated disks with uniform surface…
why can we jump

How Can We Jump When the Ground Does No Work?

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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

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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…
cpu programming

Parallel Programming on a CPU with AVX-512

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1 Comment
This article is the second of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
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Parallel Programming on an NVIDIA GPU

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1 Comment
This article is the first of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different…
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A Novel Technique of Calculating Unit Hypercube Integrals

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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

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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…
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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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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…
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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…
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Probabilistic Factors Involved in Disease and Virus Testing

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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

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176 Comments
This Insight article presents the main features of a conceptual foundation of quantum physics with the same characteristic features as classical physics…
trig special functions

A Trick to Memorizing Trig Special Angle Values Table

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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…
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How to Setup a Raspberry Pi Cluster

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9 Comments
INTRODUCTION As a long-time computer programmer and almost as long a High-Performance Computer (HPC) user, I didn't know anything about how these machines…
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An Introduction to Theorema Primum

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Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…
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Python’s Sympy Module and the Cayley-Hamilton Theorem

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Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a…
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Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

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Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief…
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Physical Applications of the “Tan Rule”

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Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,…
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Corrections to MIT Open Courseware: Systems of Varying Mass

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Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton's Laws to systems of Varying Mass PDF Course Link My concerns…
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How Quantum Information Theorists Revealed the Relativity Principle at the Foundation of Quantum Mechanics

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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.…
euler sums

Investigating Some Euler Sums

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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…
Quaternions in Projectile Motion

Quaternions in Projectile Motion

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2 Comments
Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our…
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 behave…
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When Science is Not Enough: Legal and Health Implications of Questions

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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…
LCEVC 8k tv

The LCEVC Coding Standard and 8K Television

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4 Comments
Introduction First, for background, here is an overview of the most common video compression standard in use today, H264: https://www.maketecheasier.com/how-video-compression-works/H265…
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How to Solve Projectile Motion Problems in One or Two Lines

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Introduction We show how one can solve most if not all, introductory-level projectile motion problems in one or maybe two lines. To this end, we forgo…
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10 Math Things We All Learnt Wrong At School

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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…
Fourier Series Riemann Zeta Function

Computing the Riemann Zeta Function Using Fourier Series

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5 Comments
Euler's amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous…
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Tolman Law in a Nutshell

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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…
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Maximizing Horizontal Range of a Projectile

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Introduction A recent homework problem that appeared in the forums was concerned with maximizing the horizontal range of a projectile subject to the launch…
electromagnetic computations duality

A Numerical Electromagnetic Solver Using Duality

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In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein…
Electric Field Seen by an Observer

The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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This Insight was inspired by the discussion in "electric field seen by an observer in motion", which tries to understand the relation between two expressions:…
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Valentine’s Reflections: Mathematical Matters of the Heart

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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…
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Posterior Predictive Distributions in Bayesian Statistics

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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…
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How to Use Duality in Computational Electromagnetic Problems

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1 Comment
Some weeks ago I happened across a post that caught my eye. Dale asked a question about the number of photons in an electromagnetic field. His question…
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Geometry of Side-side-angle (SSA) and Impossible Triangles

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What is the ambiguous case? In high school geometry, the idea of proofs is often first introduced to American students. A common task is to use a basic…
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Equations of Motion Revisited

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Introduction In any school Physics course, the Newtonian equations of motion are very much a 'stock' item. Students learn the equations and are given…
bayesian inference

How Bayesian Inference Works in the Context of Science

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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

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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…
Split Electric Fields

Split Electric Fields in Electrodynamics: Capacitor and Antenna

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Abstract: The analysis of the two kinds of electric fields, namely the irrotational and non-conservative, is extended to electrodynamics, as exemplified…
bayesian statistics

How to Get Started with Bayesian Statistics

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

How to Master Projectile Motion Without Quadratics

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6 Comments
Introduction In a homework thread a while back a PF member expressed dismay along the lines of "Oh no, not another boring projectile motion problem."…
electric potential

Introduction to Electric Vector Potential and Its Applications

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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?

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

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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…
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Introduction to Pitot-Static Systems in Gliders

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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?

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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

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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…
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Intro to Physically Reasonable Waves on a String

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Introduction Physics teachers who are either writing physics questions that deal with waves on a string or setting up equipment for a class lab or demo…