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

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…

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

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…

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}…

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

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…

Introduction The applicability of Newton's second law in the oft-quoted "general form"  $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$…

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…

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

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…

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…

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…

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

Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…

Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our…

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

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…

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,…

ASOE Physics — Review of Past Exam Issues ref: http://asi.edu.au/programs/past-exams/physics-olympiad-past-exams/Classification of issues: [1*]…

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

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

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…

Introduction In a previous Insight, A New Interpretation of Dr. Walter Lewin’s Paradox, I introduced the fact that there are two kinds of E fields. …

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

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…

Introduction: Maxwell's equation in differential form ## \nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}  ##  with ## \dot{\vec{E}}=0…

Introduction Preface While browsing through unanswered posts in the Classical Physics Workshop, I came across a gem at the link shown below. For the…

Definition and summary Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it? Conditions and equations $$\nabla \phi(x,y,z)…

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…

Definition/Summary The double-slit is a simple configuration used to demonstrate interference effects in waves.Equations At distances that are…

What is a limit? In mathematics, a limit describes the behavior of a function or sequence as its input approaches a particular value. Limits are a cornerstone…

In this Insights article I'll present an example that shows how Intel® AVX-512 instructions can be used to read a whole row of data in a single operation,…