The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##ddot{text{o}}##dinger noted in 1935 writing , “One can even set up quite ridiculous cases.�...
https://www.physicsforums.com/insights/schrodingers-cat-and-the-qbit/
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. For a better view, using a modern smart phone to...
https://www.physicsforums.com/insights/the-slinky-drop-experiment-analysed/
Introduction The figure on the right shows a “double-double” Atwood machine with three ideal pulleys and four masses. All pulleys are released from rest simultaneously. Which of the choic...
https://www.physicsforums.com/insights/how-to-solve-a-multi-atwood-machine-assembly/
Preamble The classical mathematician practically by instinct views the continuous process as the “real” process, and the discrete process as an approximation to it. The mathematics of finance...
https://www.physicsforums.com/insights/the-lambert-w-function-in-finance/
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 we would get a contradiction, an...
https://www.physicsforums.com/insights/why-division-by-zero-is-a-bad-idea/
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 a section on filters, but I will detail the mo...
https://www.physicsforums.com/insights/digital-filtering-and-exact-reconstruction-of-digital-audio/
Introduction First, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this article to be accessible to as wide an audience as possible,...
Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th c...
https://www.physicsforums.com/insights/series-in-mathematics-from-zeno-to-quantum-theory/
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 think I’m not the only one who struggled a...
https://www.physicsforums.com/insights/epsilontic-limits-and-continuity/
Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article, the technique has been modified and e...
https://www.physicsforums.com/insights/the-poor-mans-milli-ohm-meter/
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 sciences, especially in physics. W...
https://www.physicsforums.com/insights/differential-equation-systems-and-nature/
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 this sequence of Pre-Algebra, Algebra 1, ...
https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/
Introduction The applicability of Newton’s second law in the oft-quoted “general form” $$begin{align}frac{dmathbf{P}}{dt}=mathbf{F}_{text{ext}}end{align}$$ was an issue in a recent thread...
https://www.physicsforums.com/insights/how-to-apply-newtons-second-law-to-variable-mass-systems/
Introduction When doing mathematics, we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive. Going from rational numbers to reals is more c...
Abstract Richard Pierce describes the intention of his book about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really b...
https://www.physicsforums.com/insights/introduction-to-the-world-of-algebras/
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, Stephen Wolfram has already done the...
https://www.physicsforums.com/insights/why-chatgpt-is-not-reliable/
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) ca...
https://www.physicsforums.com/insights/what-are-infinitesimals-simple-version/
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 experiment is: a phenomenon which is i...
Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) ca...
https://www.physicsforums.com/insights/what-are-infinitesimals/
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 Lesch, Neil deGrasse Tyson, Sabine Hossenfeld...
https://www.physicsforums.com/insights/opinion-when-pro-scientists-explain-using-pop-science/
Abstract My school teacher used to say “Everybody can differentiate, but it takes an artist to integrate.” The mathematical reason behind this phrase is, that differentiation is the calculati...
https://www.physicsforums.com/insights/the-art-of-integration/
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 rationale behind this principle is arti...
https://www.physicsforums.com/insights/a-lesson-in-teaching-physics-you-cant-give-it-away/
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 that can be found in many excellent stan...
https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/
Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery – typically an AA or D cell. Typically this is based around a sim...
https://www.physicsforums.com/insights/how-to-measure-internal-resistance-of-a-battery/
Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie groups b...
https://www.physicsforums.com/insights/when-lie-groups-became-physics/
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 pressure. We will look at the two known...
https://www.physicsforums.com/insights/why-there-are-maximum-mass-limits-for-compact-objects/
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In the last article in this series, we finished up with a metric for the Oppenheimer-Snyder collapse: $$ ds^2 = – dtau^2 + A^2...
If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline what tensors are and why they are so import...
https://www.physicsforums.com/insights/what-are-tensors-and-why-are-they-used-in-relativity/
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In a previous article, I described in general terms the model of gravitational collapse of a spherically symmetric massive objec...
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 this topic, and they tend to cover the sam...
https://www.physicsforums.com/insights/when-discussing-the-twin-paradox-read-this-first/